#7623126

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1552 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,19)(13,20)(14,17)(15,29)(16,32)(18,24)(21,31)(23,38)(25,37)(26,33)(27,34)(28,39)(30,35)(36,40);(2,5)(6,8)(10,17)(18,33)(19,22)(20,32)(21,34)(29,39)(35,36)(37,38);(9,11,14,12)(10,18,17,33)(13,23,16,25)(15,30,28,40)(19,29,22,39)(20,35,32,36)(21,37,34,38)(24,27,26,31);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,19,17,22)(13,23,16,25)(15,26,28,24)(18,29,33,39)(20,38,32,37)(21,36,34,35)(27,30,31,40);(9,13,14,16)(10,20,17,32)(11,23,12,25)(15,27,28,31)(18,35,33,36)(19,37,22,38)(21,29,34,39)(24,30,26,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287046, #72320169, #72320184, …
maximal normal subgroup of	#72287046, #72320169, #72320184, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530322);