#368938

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
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

1178 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)(12,23,25,38)(13,17)(14,18)(15,21)(16,26,31,39)(20,34,35,37)(22,33)(24,32)(27,29)(28,40,30,36);(2,5)(6,7)(9,11)(10,15)(12,20,25,35)(13,21)(14,27)(16,30,31,28)(17,19)(18,33)(22,24)(23,37,38,34)(26,40,39,36)(29,32);(9,12,24,39,13,25,14,26)(10,16,18,23,17,31,32,38)(11,20,22,36,21,35,27,40)(15,28,33,34,19,30,29,37);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,31)(18,32)(23,38)(28,30)(29,33)(34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 276134);