#491403

order	512
nilpotency class	4
isoclinism	isoclinic class of #491233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,d^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=dhi,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398599);