#6770549

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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)(27,32)(28,35)(29,31)(30,34)(33,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,31,34)(26,30,29,32)(28,36,39,35)(33,38,37,40);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,31,39)(26,33,29,37)(27,35,34,36)(30,40,32,38);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,22)(21,23)(25,29,31,26)(27,30,34,32)(28,37,39,33)(35,40,36,38);(2,5)(6,8)(11,19)(14,17)(18,24)(21,23)(25,30,31,32)(26,34,29,27)(28,38,39,40)(33,35,37,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=h,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, 6677745);