#492491

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	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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 399687);