#72330646

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 7042 conjugacy classes of subgroups; only the 1105 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:

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

Polycyclic presentation:

2^10|b^2=gh,c^2=ghj,e^2=fgj,b^a=bg,c^a=cf,c^b=cgi,d^b=di,d^c=di,e^a=egij,e^b=efhj,e^c=efhj,e^d=efgj

Conjugacy classes & character table

148 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, 5, F.6*F.7*F.10);;
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, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*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*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);