#491692

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33773	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

549 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,28)(20,33)(21,26)(22,35)(23,34)(24,27)(25,30)(29,44)(31,47)(32,41)(36,39)(37,42)(38,40)(43,45)(46,48);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,22,30,24,32,21,37)(18,25,26,28,27,42,35,41)(20,29,34,43,36,47,38,46)(23,31,33,45,40,44,39,48);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,20,33,18,22,34,40,26,24,36,39,27,21,38,23,35)(19,29,45,25,30,43,44,28,32,47,48,42,37,46,31,41);(3,10)(8,15)(11,14)(12,16)(17,21,24,22)(18,26,27,35)(19,30,32,37)(20,34,36,38)(23,39,40,33)(25,41,42,28)(29,46,47,43)(31,45,44,48)

Polycyclic presentation:

2^9|b^2=egi,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, 398888);