#368982

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1180 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)(4,5)(6,8)(9,10)(11,19)(12,23,25,16)(13,17)(14,28)(15,21)(18,27)(20,33,36,29)(22,30)(24,31,39,37)(26,32,40,38)(34,35);(2,5)(6,7)(9,11)(10,15)(12,24,25,39)(13,21)(14,22)(16,31,23,37)(17,19)(18,30)(20,26,36,40)(27,35)(28,34)(29,32,33,38);(9,12,14,26,13,25,27,40)(10,16,28,38,17,23,18,32)(11,20,35,39,21,36,22,24)(15,29,30,31,19,33,34,37);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,23)(18,28)(29,33)(30,34)(31,37)(32,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,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, 276178);