#6817922

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6725118);