#7064217

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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,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, 6971413);