#492559

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	#19971	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,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, 399755);