#6719469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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:

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

Polycyclic presentation:

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

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, 6626665);