#72318425

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#5065428
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=i,b^a=bg,c^a=cgh,c^b=cf,d^a=df,d^b=dfi,d^c=di,e^a=ehj,e^b=eghi,e^c=ehj,e^d=ei,f^e=fj,g^b=gj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318468, #72318710, #72318821, …
maximal normal subgroup of	#72318468, #72318710, #72318821, …

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.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);