#369412

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1476 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)(11,19)(12,16,24,31)(13,17)(14,27)(15,21)(18,26)(20,28,37,33)(22,29)(23,38,39,30)(25,40,35,32)(34,36);(1,3,5,4)(2,6,8,7)(9,11,13,21)(10,15,17,19)(12,23)(14,22,26,36)(16,30)(18,29,27,34)(20,35)(24,39)(25,37)(28,40)(31,38)(32,33);(1,3,5,4)(2,7,8,6)(9,12,14,25,13,24,26,35)(10,16,27,40,17,31,18,32)(11,20,36,39,21,37,22,23)(15,28,29,30,19,33,34,38);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,31)(18,27)(28,33)(29,34)(30,38)(32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276608);