#7624319

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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(19,28)(20,22)(21,33)(24,37)(25,31)(27,41)(29,45)(30,44)(32,35)(34,36)(38,43)(39,48)(40,47)(42,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,23,26)(19,24,30,39)(20,31,22,25)(21,27,35,42)(28,37,44,48)(29,40,36,38)(32,46,33,41)(34,43,45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,29)(21,34)(22,36)(23,30)(25,38)(26,39)(27,43)(28,33)(31,40)(32,44)(35,45)(37,41)(42,47)(46,48);(3,11)(6,9)(10,16)(13,15)(17,20,23,22)(18,25,26,31)(19,29,30,36)(21,32,35,33)(24,38,39,40)(27,41,42,46)(28,34,44,45)(37,47,48,43);(2,8)(6,9)(7,14)(13,15)(18,26)(24,39)(25,31)(27,42)(37,48)(38,40)(41,46)(43,47)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531515);