#7006265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,23,29,20)(15,19,26,33,25,32,39,30)(16,21,28,36,27,35,40,34)(24,31,37,42,38,43,44,41);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,30)(21,31)(22,33)(23,32)(27,38)(28,37)(29,39)(34,41)(35,43)(36,42)(40,44);(5,7)(6,9)(8,12)(10,11)(14,20)(17,29)(19,30)(21,34)(22,23)(26,39)(28,40)(31,41)(32,33)(35,36)(37,44)(42,43);(1,3,4,2)(7,12)(9,11)(13,16,18,27)(14,21,23,35)(15,24,25,38)(17,28,29,40)(19,31,32,43)(20,34,22,36)(26,37,39,44)(30,41,33,42);(6,10)(9,11)(15,25)(16,27)(19,32)(21,35)(26,39)(28,40)(30,33)(34,36)

Polycyclic presentation:

2^9|a^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913461);