#7511151

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7418347);