#6376972

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 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)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,22,26,32,38)(19,23,29,35,28,34,41,45)(20,24,31,37,30,36,42,46)(27,33,40,44,39,43,47,48);(1,3,6,2)(4,7,5,8)(17,19,21,29,22,28,32,41)(18,23,25,35,26,34,38,45)(20,27,31,40,30,39,42,47)(24,33,37,44,36,43,46,48);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,39)(29,40)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6284168);