#6771512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fh,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,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6678708);