#6860315

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
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	#30722	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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)(25,26,30,33)(27,34,35,40)(28,32,29,37)(31,38,39,36);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(27,35)(31,39)(34,40)(36,38);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,39)(34,37);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(25,28,30,29)(26,32,33,37)(27,36,35,38)(31,40,39,34)

Polycyclic presentation:

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

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, 6767511);