#491601

order	512
nilpotency class	4
isoclinism	isoclinic class of #491572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

508 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:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(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);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16);(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);(3,10)(6,15)(9,16)(11,14)(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=g,b^2=e,c^2=fhi,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398797);