#468435

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

644 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:

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

Polycyclic presentation:

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

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, 375631);