#492421

order	512
nilpotency class	4
isoclinism	isoclinic class of #492236
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	#3341	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 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:

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

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=di,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399617);