#72332490

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

30451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,b^a=bj,c^b=ci,d^a=dg,d^b=dh,d^c=dgh,e^b=ef,e^c=ef

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333204, #72333205, #72333238, …
maximal normal subgroup of	#72333204, #72333205, #72333238, …

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