#491585

order	512
nilpotency class	4
isoclinism	isoclinic class of #491556
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

513 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)(19,27)(20,23)(21,34)(22,36)(24,30)(25,35)(26,33)(28,38)(29,41)(31,48)(32,39)(37,45)(40,44)(42,47)(43,46);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,23)(24,37)(25,38)(26,36)(27,41)(34,39)(40,42)(43,48);(17,19,28,44,22,31,21,30,20,29,35,47,33,46,32,45)(18,24,39,48,26,40,25,27,23,37,34,43,36,42,38,41);(3,10)(6,15)(9,16)(11,14)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fhi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398781);