#7624068

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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322060
maximal normal subgroup of	#72322060
maximal quotient of	#72343812, #72343817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531264);