#7283888

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(3,7)(17,18,23,37,22,27,19,24)(20,30,28,25,33,38,32,41)(21,34,35,39,36,26,29,42)(31,45,43,48,47,44,46,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,37)(20,28,33,32)(21,35,36,29)(25,38,41,30)(26,42,34,39)(31,46,47,43)(40,44,48,45);(10,13)(11,15)(17,20,22,33)(18,25,27,41)(19,28,23,32)(21,31,36,47)(24,38,37,30)(26,40,34,48)(29,43,35,46)(39,45,42,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,29)(20,31)(22,36)(23,35)(24,39)(25,40)(27,34)(28,43)(30,44)(32,46)(33,47)(37,42)(38,45)(41,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,27)(21,36)(24,37)(25,41)(29,35)(30,38)(31,47)(43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7191084);