#362338

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

1279 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)(9,10,14,21,15,19,13,20)(11,22,25,37,26,16,30,33)(12,17,27,35,28,18,29,36)(23,38,39,34,40,31,24,32);(9,11,13,25,15,26,14,30)(10,16,20,33,19,22,21,37)(12,24,27,40,28,39,29,23)(17,34,35,38,18,32,36,31);(2,5)(6,8)(9,12)(10,17)(11,23)(13,29)(14,27)(15,28)(16,31)(18,19)(20,36)(21,35)(22,38)(24,30)(25,39)(26,40)(32,33)(34,37);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,27)(14,29)(15,28)(16,32)(17,19)(20,36)(21,35)(22,34)(23,30)(25,40)(26,39)(31,33)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,g^2=i,h^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, 269534);