#492713

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	#1667	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^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, 399909);