#492490

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	#2206	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

409 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)(3,9,10,16)(5,13,15,8)(7,14,12,11)(17,18)(19,23)(20,24)(21,34)(22,25)(26,31)(27,32)(28,35)(29,44)(30,36)(33,39)(37,41)(38,42)(40,45)(43,46)(47,48);(1,3,6,10)(2,7,4,12)(5,14,15,11)(8,16,13,9)(17,19,22,30)(18,23,25,36)(20,28,33,43)(21,29,32,42)(24,35,39,46)(26,37,40,47)(27,38,34,44)(31,41,45,48);(3,10)(7,12)(9,16)(11,14)(18,24)(20,31)(21,32)(23,35)(25,39)(26,27)(28,41)(29,42)(33,45)(34,40)(36,46)(37,38)(43,48)(44,47);(1,4,6,2)(3,9,10,16)(5,8,15,13)(7,14,12,11)(18,25)(23,36)(24,39)(26,40)(27,34)(35,46)(37,47)(38,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,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, 399686);