#483930

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 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)(17,18,21,28,23,29,37,26)(19,30,33,45,20,34,31,25)(22,38,36,27,40,46,47,44)(24,39,41,32,43,48,42,35);(3,9)(6,12)(10,15)(13,16)(17,19)(18,24)(20,23)(21,31)(22,39)(25,44)(26,41)(27,45)(28,42)(29,43)(30,46)(32,47)(33,37)(34,38)(35,36)(40,48);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,20,23,19)(18,25)(21,31,37,33)(22,32,40,35)(24,27)(26,34)(28,30)(29,45)(36,48,47,39)(38,42)(41,46)(43,44);(2,7)(6,12)(8,14)(13,16)(17,21,23,37)(18,26,29,28)(19,31,20,33)(22,36,40,47)(24,41,43,42)(25,30,45,34)(27,38,44,46)(32,39,35,48)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391126);