#363490

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 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)(4,6)(5,8)(9,10,16,20,15,21,14,31)(11,19,13,30,24,38,23,17)(12,25,28,36,29,18,32,37)(22,39,40,34,26,35,27,33);(1,3,4,8)(2,5,6,7)(9,11,14,23,15,24,16,13)(10,17,20,19,21,30,31,38)(12,26,28,27,29,22,32,40)(18,34,36,39,25,33,37,35);(3,8)(5,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,29)(16,32)(17,33)(19,35)(20,36)(21,25)(23,40)(24,26)(30,34)(31,37)(38,39);(2,6)(5,7)(9,13,16,24,15,23,14,11)(10,19,31,17,21,38,20,30)(12,27,32,26,29,40,28,22)(18,35,37,33,25,39,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eg,e^2=i,h^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, 270686);