#457137

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1135 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=i,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=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, 364333);