#479939

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bi,d^a=dg,c^b=cgi,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387135);