#6758047

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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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)(3,7)(4,6)(5,8)(17,18,24,38,23,29,22,28)(19,25,31,41,33,43,32,42)(20,34,36,26,21,27,37,45)(30,40,47,39,35,48,46,44);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,25,29,43)(20,35,21,30)(22,32,24,31)(26,44,45,39)(27,40,34,48)(28,42,38,41)(36,46,37,47);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,45)(19,30,33,35)(22,36,24,37)(25,39,43,44)(27,38,34,28)(31,46,32,47)(40,42,48,41);(2,6)(5,7)(17,21,23,20)(18,27,29,34)(19,30,33,35)(22,36,24,37)(25,40,43,48)(26,38,45,28)(31,47,32,46)(39,42,44,41);(10,14)(13,15)(17,22,23,24)(18,28,29,38)(19,31,33,32)(20,36,21,37)(25,41,43,42)(26,27,45,34)(30,46,35,47)(39,40,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fh,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6665243);