#496763

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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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)(3,7)(4,6)(5,8)(9,10,12,19,13,20,26,35,14,18,25,34,27,17,32,33)(11,15,22,29,23,28,37,39,21,16,24,31,36,30,38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,23)(14,21)(16,18)(17,30)(19,29)(20,28)(24,25)(26,37)(27,36)(31,34)(32,38)(33,40)(35,39);(3,8)(5,7)(9,11)(10,16)(12,24)(13,23)(14,21)(15,18)(17,28)(19,29)(20,30)(22,25)(26,38)(27,36)(31,34)(32,37)(33,40)(35,39);(2,6)(5,7)(10,17,18,20)(11,21)(12,25)(15,28,16,30)(19,35,34,33)(23,36)(26,32)(29,40,31,39)

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^b=di,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, 403959);