#392542

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 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)(4,8)(5,6)(9,10)(11,22)(12,25,28,34)(13,16)(14,20)(15,21)(17,27,35,40)(18,24)(19,30)(23,31,37,38)(26,33,39,36)(29,32);(2,6)(7,8)(9,11,14,13,15,24,30,29)(10,16,19,22,21,32,20,18)(12,23,27,26,28,37,40,39)(17,31,34,36,35,38,25,33);(1,3,5,4)(2,7,6,8)(9,12,15,28)(10,17)(11,23,24,37)(13,26,29,39)(14,27,30,40)(16,31)(18,33)(19,34)(20,25)(21,35)(22,36)(32,38);(1,4,5,3)(2,8,6,7)(9,13,14,24,15,29,30,11)(10,18,19,16,21,22,20,32)(12,26,27,37,28,39,40,23)(17,33,34,31,35,36,25,38)

Polycyclic presentation:

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

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, 299738);