#72285755

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3355 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,12)(5,14)(6,9)(7,10)(11,13)(15,16)(20,28)(23,29);(1,3)(2,8)(4,10)(5,15,6,11)(7,12)(9,13,14,16)(17,18)(19,25)(20,29,28,23)(21,24)(22,27)(26,32,30,31);(1,4,7,8)(2,3,10,12)(5,13)(6,16)(9,15)(11,14)(17,19)(18,22)(20,30,28,26)(21,27)(23,32,29,31)(24,25);(1,5,7,6)(2,9)(3,11,12,15)(4,13)(8,16)(10,14)(17,19)(18,22,24,25)(20,26)(21,27)(23,31,29,32)(28,30);(1,6,7,5)(2,9)(3,11,12,15)(4,13)(8,16)(10,14)(17,20)(18,23,24,29)(19,26)(21,28)(22,31,25,32)(27,30)

Polycyclic presentation:

2^10|b^2=f,c^2=fi,d^2=gi,e^2=gi,c^a=cfi,c^b=cfj,d^a=df,d^b=dh,d^c=dfg,e^a=efhij,e^b=eghij,e^c=efhi,f^d=fi,f^e=fij,g^a=gi,g^b=gj,g^c=gi,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72311321, #72338686, …
maximal normal subgroup of	#72310969, #72311321, #72338686, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);