#483437

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ch,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390633);