#492495

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

401 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)(17,18,23,26)(19,24,22,37)(20,30,32,47)(21,33,36,46)(25,40,41,31)(27,29,43,35)(28,45,38,44)(34,39,48,42);(1,3,6,10)(2,7,4,12)(5,14,15,11)(8,16,13,9)(17,19,23,22)(18,24,26,37)(20,28,32,38)(21,29,36,35)(25,39,41,42)(27,33,43,46)(30,45,47,44)(31,34,40,48);(3,10)(7,12)(9,16)(11,14)(17,20,23,32)(18,25,26,41)(19,28,22,38)(21,34,36,48)(24,39,37,42)(27,44,43,45)(29,40,35,31)(30,46,47,33);(1,4,6,2)(3,9,10,16)(5,8,15,13)(7,14,12,11)(18,26)(24,37)(25,41)(27,43)(30,47)(33,46)(39,42)(44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,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, 399691);