#468935

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=gi,c^2=g,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376131);