#6775067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(3,6)(4,5)(7,8)(17,18,21,25)(19,27,29,23)(20,31,33,42)(22,36,35,26)(24,37,41,34)(28,44,32,48)(30,39,46,43)(38,45,40,47);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,39)(28,33)(31,44)(34,47)(35,46)(36,43)(37,45)(38,41)(42,48);(10,13)(14,16)(17,20,21,33)(18,24,25,41)(19,28,29,32)(22,37,35,34)(23,38,27,40)(26,42,36,31)(30,47,46,45)(39,44,43,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(24,41)(26,36)(28,32)(30,46)(31,42)(45,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(23,27)(28,32)(30,46)(38,40)(39,43)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682263);