#6393298

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1566 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)(11,13)(12,14)(15,16)(17,18,21,26,22,25,32,38)(19,23,29,35,28,34,41,45)(20,24,31,37,30,36,42,46)(27,33,39,44,40,43,47,48);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,34)(26,35)(30,40)(31,39)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(10,14)(13,16)(17,20,21,31,22,30,32,42)(18,24,26,37,25,36,38,46)(19,27,29,39,28,40,41,47)(23,33,35,44,34,43,45,48);(1,4,6,5)(2,7,3,8)(18,25)(23,34)(24,36)(26,38)(33,43)(35,45)(37,46)(44,48);(1,5,6,4)(2,7,3,8)(19,28)(20,30)(23,34)(24,36)(29,41)(31,42)(35,45)(37,46)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei,e^c=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, 6300494);