#6486579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,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, 6393775);