#7624228

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1511 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,20,19,16)(12,23,25,38)(15,30,29,22)(17,34,24,26)(18,35,28,32)(21,31,37,39)(27,40,33,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,20)(14,19)(15,22)(17,33)(18,32)(23,39)(24,27)(25,37)(26,36)(28,35)(29,30)(31,38)(34,40);(9,12,14,25)(10,17,13,24)(11,21,19,37)(15,31,29,39)(16,33,20,27)(18,36,28,40)(22,38,30,23)(26,35,34,32);(3,8)(5,7)(9,13,14,10)(11,20,19,16)(12,24,25,17)(15,28,29,18)(21,27,37,33)(22,35,30,32)(23,26,38,34)(31,40,39,36);(2,6)(5,7)(10,13)(11,19)(17,24)(18,28)(21,37)(22,30)(23,38)(36,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^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=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei

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, 7531424);