#362336

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
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

1315 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)(3,6)(4,5)(7,8)(9,10)(11,19,23,15)(12,24,26,16)(13,18)(14,17)(20,30,22,28)(21,29)(25,37,38,31)(27,32)(33,36)(34,39)(35,40);(9,11,13,22,14,23,27,20)(10,15,18,30,17,19,32,28)(12,21,25,35,26,36,38,34)(16,29,31,40,24,33,37,39);(2,5)(6,8)(11,20)(13,27)(15,28)(18,32)(19,30)(21,34)(22,23)(25,38)(29,39)(31,37)(33,40)(35,36);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,21)(13,25)(14,26)(15,29)(17,24)(18,31)(19,33)(20,34)(22,35)(23,36)(27,38)(28,39)(30,40)(32,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,h^d=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, 269532);