#6770563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6677759);