#483866

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	C₂×C₄×D₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 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)(3,6,9,13,10,12,15,16)(17,18,22,28)(19,25,31,44)(20,26,23,38)(21,34,36,48)(24,39,41,30)(27,33,46,37)(29,43,32,45)(35,40,47,42);(17,19)(18,24)(20,29)(21,35)(22,31)(23,32)(25,34)(26,40)(27,45)(28,41)(30,37)(33,39)(36,47)(38,42)(43,46)(44,48);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25)(20,32)(21,30)(22,31)(23,29)(24,34)(26,45)(27,40)(28,44)(33,47)(35,37)(36,39)(38,43)(41,48)(42,46);(2,7)(6,12)(8,14)(13,16)(17,20,22,23)(18,26,28,38)(19,29,31,32)(21,33,36,37)(24,40,41,42)(25,43,44,45)(27,34,46,48)(30,35,39,47)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,e^b=eg,g^b=gi,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, 391062);