#6668094

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2706 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575290);