#6378978

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1294 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)(11,16)(12,14)(13,15)(17,18,21,25,22,26,33,39)(19,23,29,36,27,34,41,45)(20,30,32,44,31,24,37,38)(28,42,43,48,40,35,46,47);(1,3,4,8)(2,5,6,7)(19,27)(23,34)(28,40)(29,41)(35,42)(36,45)(43,46)(47,48);(3,8)(5,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,35)(25,36)(26,34)(30,42)(31,40)(32,43)(33,41)(37,46)(38,47)(39,45)(44,48);(2,6)(5,7)(19,27)(20,31)(23,34)(24,30)(29,41)(32,37)(36,45)(38,44);(9,12,13,11)(10,14,15,16)(17,20,21,32,22,31,33,37)(18,24,25,38,26,30,39,44)(19,28,29,43,27,40,41,46)(23,35,36,47,34,42,45,48)

Polycyclic presentation:

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

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, 6286174);