#6469329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,27)(20,25)(21,33)(22,24)(23,30)(26,34)(28,40)(29,39)(31,37)(32,45)(35,43)(36,42)(38,44)(41,47)(46,48);(9,10)(11,16)(12,14)(13,15)(17,19,21,30,22,28,34,42)(18,23,26,27,24,36,33,40)(20,29,32,43,31,41,44,48)(25,35,38,39,37,46,45,47);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,24)(19,28)(25,37)(26,33)(29,41)(30,42)(38,45)(43,48);(10,14)(13,16)(17,20)(18,25)(19,29)(21,32)(22,31)(23,35)(24,37)(26,38)(27,39)(28,41)(30,43)(33,45)(34,44)(36,46)(40,47)(42,48);(1,4,5,3)(2,6,7,8)(20,31)(25,37)(29,41)(32,44)(35,46)(38,45)(39,47)(43,48)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376525);