#7511163

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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

1654 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)(19,23)(20,24)(21,26)(22,27)(25,29)(28,37)(30,38)(31,39)(32,41)(33,42)(34,43)(35,44)(36,45)(40,46)(47,48);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,31)(21,34)(22,29)(24,39)(25,27)(26,43)(28,32)(30,36)(33,47)(35,46)(37,41)(38,45)(40,44)(42,48);(10,14)(13,15)(17,20,22,33)(18,24,27,42)(19,28,29,46)(21,35,36,32)(23,37,25,40)(26,44,45,41)(30,47,34,31)(38,48,43,39);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(17,19)(18,25)(20,28)(21,30)(22,29)(23,27)(24,40)(26,43)(31,32)(33,46)(34,36)(35,47)(37,42)(38,45)(39,44)(41,48);(19,29)(23,25)(28,46)(30,34)(31,47)(37,40)(38,43)(39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7418359);