#7624652

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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

1348 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)(3,7)(9,10)(11,17)(12,25)(13,20)(14,29)(15,19)(16,22)(18,27)(21,31)(23,34)(24,37)(26,33)(28,39)(30,36)(32,38)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11,16,12)(10,17,22,25)(13,26,15,23)(14,24,32,27)(18,29,37,38)(19,34,20,33)(21,35,39,36)(28,30,31,40);(9,12,16,11)(10,18,22,37)(13,26,15,23)(14,30,32,40)(17,29,25,38)(19,35,20,36)(21,34,39,33)(24,31,27,28);(2,5)(6,8)(9,13,16,15)(10,19,22,20)(11,23,12,26)(14,28,32,31)(17,33,25,34)(18,35,37,36)(21,38,39,29)(24,40,27,30);(1,3,4,7)(2,6,5,8)(9,13,16,15)(10,20,22,19)(11,23,12,26)(14,28,32,31)(17,34,25,33)(18,36,37,35)(21,29,39,38)(24,40,27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318797, #72320442, #72320497
maximal normal subgroup of	#72318797, #72320442, #72320497

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531848);