#6855766

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6762962);