#364182

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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

1460 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)(11,18)(12,25,29,17)(13,30)(14,31)(15,20)(16,23)(19,21)(22,38,26,34)(24,36)(27,32,39,33)(28,35,37,40);(1,3)(2,5)(4,8)(6,7)(9,11,21,13,15,24,14,23)(10,16,31,36,20,30,19,18)(12,26,37,39,29,22,28,27)(17,33,40,34,25,32,35,38);(3,8)(5,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,32)(18,34)(19,35)(20,25)(21,37)(23,39)(24,26)(30,33)(31,40)(36,38);(2,6)(5,7)(9,13,21,24,15,23,14,11)(10,18,31,16,20,36,19,30)(12,27,37,26,29,39,28,22)(17,34,40,32,25,38,35,33)

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,d^b=di,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, 271378);