#6832021

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(20,32)(24,39)(29,43)(33,46)(34,40)(37,44)(41,48)(45,47);(1,3)(2,5)(4,8)(6,7)(17,18,22,26)(19,27,31,42)(20,33,32,46)(21,35,28,25)(23,36,38,30)(24,40,39,34)(29,44,43,37)(41,47,48,45);(10,13)(14,16)(17,19,28,36,22,31,21,30)(18,23,35,42,26,38,25,27)(20,29,40,47,32,43,34,45)(24,37,46,48,39,44,33,41);(2,6)(5,7)(20,32)(24,39)(29,43)(33,46)(34,40)(37,44)(41,48)(45,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,34,22,32,28,40)(18,24,25,33,26,39,35,46)(19,29,30,45,31,43,36,47)(23,37,27,41,38,44,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739217);