#391278

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1057 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)(3,5)(4,6)(7,8)(9,10)(11,21,24,36)(12,25)(13,18)(14,19)(15,20)(16,23,33,31)(17,28)(22,38,26,34)(27,35)(29,32,39,40)(30,37);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,26)(13,29)(14,31)(15,24)(17,32)(18,34)(19,36)(20,33)(21,37)(22,28)(23,30)(25,40)(27,39)(35,38);(2,6)(5,8)(9,12,15,28)(10,17,20,25)(11,22,24,26)(13,30,27,14)(16,32,33,40)(18,19,35,37)(21,34,36,38)(23,29,31,39);(3,7)(5,8)(9,13,14,12,15,27,30,28)(10,18,19,25,20,35,37,17)(11,22,23,39,24,26,31,29)(16,32,21,38,33,40,36,34)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298474);