#454366

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1865 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)(3,4)(6,7)(9,10,14,19)(11,22,25,17)(12,28,18,33)(13,31,32,30)(15,21,16,20)(23,24,39,40)(26,36,27,37)(29,34,35,38);(5,6,8,7)(9,11)(10,17)(12,29)(13,24)(14,25)(15,27)(16,26)(18,35)(19,22)(20,37)(21,36)(23,31)(28,38)(30,39)(32,40)(33,34);(2,4)(6,7)(9,12,14,18)(10,13,19,32)(11,23,25,39)(15,31,16,30)(17,34,22,38)(20,33,21,28)(24,37,40,36)(26,29,27,35);(5,7,8,6)(9,13,14,32)(10,18,19,12)(11,24,25,40)(15,28,16,33)(17,35,22,29)(20,31,21,30)(23,36,39,37)(26,34,27,38)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,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, 361562);