#6432792

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(9,10)(11,14)(12,13)(15,16)(17,18)(19,30)(20,35)(21,26)(22,27)(23,28)(24,29)(25,34)(31,40)(32,47)(33,45)(36,41)(37,44)(38,43)(39,42)(46,48);(2,5)(6,8)(17,19,23,34)(18,25,28,30)(20,36,21,32)(22,33,24,39)(26,41,35,47)(27,42,29,45)(31,38,48,37)(40,44,46,43);(10,13)(14,16)(17,20,23,21)(18,26,28,35)(19,31,34,48)(22,37,24,38)(25,40,30,46)(27,43,29,44)(32,39,36,33)(41,42,47,45);(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,26,28,35)(19,32,34,36)(22,37,24,38)(25,41,30,47)(27,44,29,43)(31,39,48,33)(40,42,46,45);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,27,28,29)(19,33,34,39)(20,37,21,38)(25,42,30,45)(26,43,35,44)(31,32,48,36)(40,47,46,41)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339988);