#489136

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396332);