#7509460

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,25)(19,23,30,36)(20,24,31,39)(21,26,33,40)(27,34,29,37)(28,35,42,47)(32,38,44,48)(41,45,43,46);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,27)(22,30)(24,35)(25,36)(26,34)(29,33)(31,42)(32,41)(37,40)(38,45)(39,47)(43,44)(46,48);(9,12,13,11)(10,14,15,16)(19,27)(21,33)(23,34)(26,40)(28,41)(29,30)(32,44)(35,45)(36,37)(38,48)(42,43)(46,47);(2,7)(3,8)(10,15)(14,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,39)(26,38)(27,41)(29,43)(30,42)(33,44)(34,45)(36,47)(37,46)(40,48);(1,4,5,6)(2,3,7,8)(18,25)(20,31)(23,36)(26,40)(28,42)(32,44)(34,37)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7416656);