#7624046

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#61485	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1282 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531242);