#6383182

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290378);