#484007

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
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 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

423 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)(17,18,21,27,24,29,23,39)(19,30,32,41,34,47,40,46)(20,26,35,45,36,28,22,37)(25,38,42,31,44,48,43,33);(17,19)(18,25)(20,31)(21,32)(22,38)(23,40)(24,34)(26,41)(27,42)(28,46)(29,44)(30,37)(33,36)(35,48)(39,43)(45,47);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,23,22,24,36,21,35)(18,26,39,37,29,28,27,45)(19,31,40,38,34,33,32,48)(25,41,43,30,44,46,42,47);(3,9)(6,12)(10,15)(13,16)(17,21,24,23)(18,27,29,39)(19,32,34,40)(20,35,36,22)(25,42,44,43)(26,45,28,37)(30,41,47,46)(31,48,33,38)

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,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391203);