#7623130

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
prime radical	#33522	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1534 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)(11,21)(12,16)(13,18)(14,17)(15,29)(19,28)(20,27)(22,35)(23,32)(24,30)(25,37)(26,33)(31,39)(34,40)(36,38);(2,5)(6,8)(9,11,13,12)(10,16,18,21)(14,22,26,24)(15,23,27,25)(17,30,33,35)(19,31,34,36)(20,32,29,37)(28,38,40,39);(10,17)(14,28)(15,27)(16,30)(18,33)(19,20)(21,35)(22,38)(23,25)(24,39)(26,40)(29,34)(31,32)(36,37);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,18,21)(14,24,26,22)(15,25,27,23)(17,30,33,35)(19,31,34,36)(20,32,29,37)(28,39,40,38);(11,12)(16,21)(22,24)(23,25)(30,35)(31,36)(32,37)(38,39)

Polycyclic presentation:

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

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, 7530326);