#6520215

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 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)(13,14,18,24,19,21,17,23)(15,25,27,34,28,38,33,20)(16,29,30,37,31,22,32,36)(26,40,42,35,43,39,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,21,34)(16,26,31,43)(17,27,18,33)(22,35,29,44)(23,25,24,38)(30,41,32,42)(36,40,37,39);(1,2,3,4)(6,9)(7,11)(14,21)(20,34)(22,29)(23,24)(25,38)(35,44)(36,37)(39,40);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,32)(18,30)(19,31)(20,35)(21,29)(23,37)(24,36)(25,39)(27,42)(28,43)(33,41)(34,44)(38,40);(6,9)(10,12)(13,17,19,18)(14,23,21,24)(15,27,28,33)(16,30,31,32)(20,25,34,38)(22,36,29,37)(26,41,43,42)(35,40,44,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427411);