#6500369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
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	C₂×#3197	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2693 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407565);