#7624318

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

1238 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)(17,18,21,26)(19,29,31,46)(20,28,23,25)(22,35,34,42)(24,39,41,32)(27,33,43,37)(30,47,38,44)(36,45,48,40);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,40)(31,41)(33,42)(34,43)(35,37)(36,44)(38,45)(39,46)(47,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,31)(18,24,26,41)(20,30,23,38)(22,36,34,48)(25,40,28,45)(27,44,43,47)(29,42,46,35)(32,33,39,37);(3,11)(6,9)(10,16)(13,15)(17,20,21,23)(18,25,26,28)(19,30,31,38)(22,33,34,37)(24,40,41,45)(27,42,43,35)(29,44,46,47)(32,36,39,48);(2,8)(6,9)(7,14)(13,15)(18,26)(24,41)(25,28)(27,43)(29,46)(35,42)(40,45)(44,47)

Polycyclic presentation:

2^9|a^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, 7531514);