#6820127

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,25)(21,26)(22,37)(23,28)(27,30)(29,38)(31,39)(32,40)(33,41)(34,42)(35,46)(36,43)(44,47)(45,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,32)(21,22)(23,31)(25,40)(26,37)(27,41)(28,39)(29,34)(30,33)(35,36)(38,42)(43,46)(44,45)(47,48);(9,12,13,11)(10,14,15,16)(17,20,33,45,23,36,21,34)(18,25,41,48,28,43,26,42)(19,29,22,35,31,44,30,32)(24,38,37,46,39,47,27,40);(2,6)(5,7)(17,21,23,33)(18,26,28,41)(19,22,31,30)(20,34,36,45)(24,37,39,27)(25,42,43,48)(29,35,44,32)(38,46,47,40);(3,8)(5,7)(17,22,33,31,23,30,21,19)(18,27,41,24,28,37,26,39)(20,35,45,44,36,32,34,29)(25,40,48,38,43,46,42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6727323);