#6820871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6728067);