#422675

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1085 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329871);