#489159

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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

663 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)(18,25)(19,30)(20,23)(21,35)(22,34)(27,36)(28,38)(29,41)(31,37)(32,39)(33,48)(40,44)(42,45)(43,47);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18)(19,26)(20,28)(21,36)(22,38)(23,27)(24,29)(25,35)(30,47)(31,45)(32,42)(33,46)(34,41)(37,44)(39,43)(40,48);(17,19)(18,26)(20,31)(21,37)(22,39)(23,32)(24,33)(25,40)(27,42)(28,45)(29,46)(30,34)(35,48)(36,44)(38,43)(41,47);(17,20,24,23)(18,27,29,28)(19,31,33,32)(21,34,22,35)(25,38,41,36)(26,42,46,45)(30,39,48,37)(40,43,47,44)

Polycyclic presentation:

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

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, 396355);