#6432800

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 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)(19,25)(20,26)(21,35)(22,37)(23,27)(24,28)(29,39)(30,47)(31,42)(32,43)(33,40)(34,45)(36,44)(38,41)(46,48);(2,5)(6,8)(17,19,24,32)(18,25,28,43)(20,33,21,30)(22,31,23,38)(26,40,35,47)(27,41,37,42)(29,36,46,34)(39,44,48,45);(10,13)(14,16)(17,20,24,21)(18,26,28,35)(19,29,32,46)(22,34,23,36)(25,39,43,48)(27,44,37,45)(30,38,33,31)(40,42,47,41);(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,26,28,35)(19,30,32,33)(22,34,23,36)(25,40,43,47)(27,45,37,44)(29,38,46,31)(39,42,48,41);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,27,28,37)(19,31,32,38)(20,34,21,36)(25,41,43,42)(26,44,35,45)(29,30,46,33)(39,40,48,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6339996);