#6458872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,25)(21,34)(22,24)(23,30)(26,35)(28,42)(29,41)(31,46)(32,38)(33,48)(36,45)(37,43)(39,47)(40,44);(9,11)(10,13)(12,15)(14,16)(17,19,21,30,22,28,35,43)(18,23,26,27,24,37,34,42)(20,31,33,47,32,29,44,45)(25,39,40,46,38,36,48,41);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,24)(19,28)(25,38)(26,34)(29,31)(30,43)(40,48)(45,47);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,32)(18,25,24,38)(19,29,28,31)(21,33,35,44)(23,36,37,39)(26,40,34,48)(27,41,42,46)(30,45,43,47);(20,32)(25,38)(29,31)(33,44)(36,39)(40,48)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6366068);