#72330104

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	#156684	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6994 conjugacy classes of subgroups; only the 1067 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=gh,c^2=ghj,d^2=fgh,b^a=bg,c^a=cf,c^b=cgi,d^a=dfgij,d^b=dfgij,d^c=dfgi,e^a=efgij,e^b=efhj,e^c=efh,e^d=efgh

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

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