#491669

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

510 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398865);