#491670

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	#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

527 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)(19,29)(20,34)(21,26)(22,39)(23,38)(24,28)(25,31)(27,35)(30,32)(33,41)(36,40)(37,43)(42,47)(44,45)(46,48);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,21,37,24,33,22,31)(18,25,39,41,28,43,26,29)(20,30,38,44,27,42,36,46)(23,32,34,48,40,47,35,45);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,35,28,22,36,40,39,24,27,34,18,21,38,23,26)(19,30,45,43,31,46,47,41,33,42,48,25,37,44,32,29);(3,9)(7,14)(10,15)(12,16)(17,21,24,22)(18,26,28,39)(19,31,33,37)(20,36,27,38)(23,34,40,35)(25,41,43,29)(30,44,42,46)(32,45,47,48)

Polycyclic presentation:

2^9|b^2=ei,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, 398866);