#6865696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#31001	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(17,18,22,26)(19,28,30,24)(20,33,35,41)(21,27,23,36)(25,37,40,34)(29,44,45,48)(31,43,32,39)(38,46,42,47);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,24,26,28)(20,29,35,45)(21,31,23,32)(25,38,40,42)(27,39,36,43)(33,48,41,44)(34,46,37,47);(11,16)(13,15)(17,20,22,35)(18,25,26,40)(19,29,30,45)(21,34,23,37)(24,38,28,42)(27,41,36,33)(31,46,32,47)(39,44,43,48);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(18,26)(19,30)(25,40)(27,36)(29,45)(31,32)(33,41)(46,47);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,27,26,36)(19,31,30,32)(20,34,35,37)(24,39,28,43)(25,41,40,33)(29,46,45,47)(38,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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,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, 6772892);