#6468639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1558 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)(10,14)(15,16)(17,18,23,36,22,27,21,26)(19,28,35,24,30,43,31,39)(20,25,37,48,33,41,34,42)(29,44,47,38,32,46,45,40);(9,10,13,14)(11,16,12,15)(17,19,22,30)(18,24,27,39)(20,32,33,29)(21,35,23,31)(25,40,41,38)(26,43,36,28)(34,45,37,47)(42,44,48,46);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,33)(18,25,27,41)(19,29,30,32)(21,34,23,37)(24,38,39,40)(26,42,36,48)(28,44,43,46)(31,45,35,47);(9,12,13,11)(10,16,14,15)(19,30)(24,39)(28,43)(29,32)(31,35)(38,40)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(20,33)(25,41)(29,32)(34,37)(38,40)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375835);