#72284366

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3327 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|d^2=gh,e^2=ghj,c^a=cf,c^b=cfj,d^a=df,d^c=dfg,e^a=efhij,e^b=eghj,e^c=ef,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^d=gj,g^e=gj,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	#72284848, #72285190, #72310982, …
maximal normal subgroup of	#72284848, #72285190, #72310982, …

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, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
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.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
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.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
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);