#492788

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	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×D₁₆	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 399984);