#495445

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	C₂³	C₂²×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

420 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)(3,6)(4,8)(5,7)(9,10,15,20,14,21,34,39,16,22,36,18,35,19,13,32)(11,23,26,29,27,31,37,38,28,40,24,33,25,12,17,30);(1,3,5,4)(2,6,7,8)(9,11,15,26,14,27,34,37,16,28,36,24,35,25,13,17)(10,12,20,30,21,23,39,29,22,31,18,38,19,40,32,33);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,20)(13,33)(14,23)(15,30)(16,31)(18,28)(19,24)(21,26)(22,37)(25,32)(27,39)(29,34)(35,40)(36,38);(9,13,14,15,16,34,35,36)(10,18,19,39,22,20,21,32)(11,24,25,37,28,26,27,17)(12,29,23,38,31,33,40,30)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^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,d^b=dh,e^b=ei,d^c=di,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, 402641);