#7138680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7045876);