#6865783

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,40)(29,43)(33,39)(36,37)(38,48)(41,42)(44,45)(46,47);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(18,23)(19,28)(25,31)(26,35)(29,42)(30,36)(39,46)(44,48);(11,16)(13,15)(17,19,21,28)(18,24,23,37)(20,29,32,42)(22,36,34,30)(25,38,31,45)(26,40,35,27)(33,48,47,44)(39,43,46,41);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20,21,32)(18,25,23,31)(19,29,28,42)(22,33,34,47)(24,38,37,45)(26,39,35,46)(27,41,40,43)(30,44,36,48);(1,5,6,4)(2,7,3,8)(20,32)(25,31)(29,42)(33,47)(38,45)(39,46)(41,43)(44,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

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, 6772979);