#7504855

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,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,e^d=ei

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, 7412051);