#494134

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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)(3,6)(4,5)(7,8)(9,10,14,20,13,21,29,36,15,17,31,18,30,19,12,27)(11,22,25,37,24,16,23,33,26,34,38,40,39,32,28,35);(2,5)(6,8)(9,11)(10,16,17,32)(12,28)(13,24)(14,25)(15,26)(18,35,20,33)(19,22,21,34)(23,29)(27,37,36,40)(30,39)(31,38);(10,17)(12,29)(14,31)(16,32)(19,21)(22,34)(23,28)(25,38);(1,3,4,7)(2,6,5,8)(9,12,13,14,15,29,30,31)(10,18,19,36,17,20,21,27)(11,23,24,38,26,28,39,25)(16,33,22,37,32,35,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401330);