#6757825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6665021);