#6577823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

1964 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)(3,4)(5,6)(7,11)(8,12)(9,10);(6,10)(11,12)(13,14,18,22)(15,23,26,36)(16,27,31,20)(17,32,34,21)(19,33,37,25)(24,41,43,38)(28,44,42,35)(29,40,30,39);(2,4)(13,15,18,26)(14,19,22,37)(16,28,31,42)(17,33,34,25)(20,38,27,41)(21,36,32,23)(24,30,43,29)(35,39,44,40);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,29)(18,31)(19,35)(21,40)(22,27)(23,38)(25,28)(26,43)(30,34)(32,39)(33,42)(36,41)(37,44);(5,8,9,7)(6,12,10,11)(14,21)(16,29)(17,34)(19,36)(20,27)(22,32)(23,37)(24,28)(25,33)(30,31)(38,41)(42,43)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485019);