#6850916

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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)(17,18,22,24)(19,28,31,40)(20,32,34,26)(21,27,23,36)(25,37,38,30)(29,44,46,43)(33,42,35,41)(39,47,48,45);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(18,24)(25,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(9,11,13,12)(10,14,16,15)(17,19)(18,25)(20,29)(21,30)(22,31)(23,37)(24,38)(26,39)(27,40)(28,36)(32,48)(33,45)(34,46)(35,47)(41,43)(42,44);(2,6)(5,7)(17,20,22,34)(18,26,24,32)(19,29,31,46)(21,35,23,33)(25,39,38,48)(27,42,36,41)(28,43,40,44)(30,47,37,45);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,33,34,35)(25,40,38,28)(26,41,32,42)(29,45,46,47)(39,43,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6758112);