#7062435

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6969631);