#7624911

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 7532107);