#492790

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399986);