#496761

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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)(3,7)(4,6)(5,8)(9,10,12,18,13,19,26,35,14,17,25,33,27,16,30,32)(11,20,22,34,23,31,37,40,21,15,24,29,36,28,38,39);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11)(10,15)(12,24)(13,23)(14,21)(16,31)(17,20)(18,34)(19,28)(22,25)(26,38)(27,36)(29,33)(30,37)(32,39)(35,40);(2,6)(5,7)(10,16,17,19)(11,21)(12,25)(15,28,20,31)(18,35,33,32)(23,36)(26,30)(29,40,34,39)

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=ce,d^c=di,g^c=gi,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403957);