#364181

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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×#380
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 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)(12,24,27,16)(13,28)(14,18)(15,22)(17,20)(21,37,25,33)(23,34)(26,32,36,39)(29,38,31,40)(30,35);(1,3)(2,5)(4,8)(6,7)(9,11,20,35,14,23,13,22)(10,15,28,34,18,30,17,19)(12,25,36,38,27,21,26,40)(16,31,39,33,24,29,32,37);(3,8)(5,7)(9,12)(10,16)(11,21)(13,26)(14,27)(15,29)(17,32)(18,24)(19,33)(20,36)(22,38)(23,25)(28,39)(30,31)(34,37)(35,40);(2,6)(5,7)(9,11,20,35,14,23,13,22)(10,15,28,34,18,30,17,19)(12,21,36,40,27,25,26,38)(16,29,39,37,24,31,32,33)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=eg,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, 271377);