#72323689

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂×#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

30187 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fgj,e^2=fg,b^a=bg,c^a=ch,d^a=dfh,e^b=ef,e^c=ehi,e^d=efghi

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323738, #72323741, #72323769
maximal normal subgroup of	#72323738, #72323741, #72323769

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