#362966

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×#380
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,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, 270162);