#467485

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,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, 374681);