#72329957

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

No aliases for this group.

Subgroups

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

35019 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,e^2=g,b^a=bj,c^b=cf,d^a=dh,d^b=dh,e^a=efg,e^b=efgh,e^d=ei

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331079, #72331105, #72331219, …
maximal normal subgroup of	#72331079, #72331105, #72331219, …

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