#6490337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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

1383 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,g^c=gi,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, 6397533);