#7623767

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343987
maximal normal subgroup of	#72343987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530963);