#7624345

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1247 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)(19,25)(20,26)(21,24)(22,35)(23,29)(27,34)(28,33)(30,39)(31,38)(32,40)(36,44)(37,48)(41,47)(42,46)(43,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,31)(24,36)(25,26)(27,37)(28,38)(32,44)(33,45)(34,46)(35,43)(39,40)(41,42)(47,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(21,34)(22,33)(23,36)(25,39)(26,40)(27,28)(30,46)(31,45)(32,47)(35,41)(37,38)(42,43)(44,48);(3,11)(6,9)(10,16)(13,15)(17,19)(18,25)(20,29)(21,30)(22,31)(23,26)(24,39)(27,42)(28,43)(32,44)(33,45)(34,46)(35,38)(36,40)(37,41)(47,48);(2,8)(6,9)(7,14)(13,15)(18,26)(19,29)(24,40)(27,41)(28,35)(30,44)(31,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531541);