#372532

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 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)(3,8)(4,6)(5,7)(9,10,16,19)(11,14,22,15)(12,20)(13,21,30,35)(17,25)(18,29,33,28)(23,27)(24,37)(26,32)(31,40)(34,38)(36,39);(1,3,5,4)(2,6,7,8)(9,11,14,19,16,22,15,10)(12,23,26,32,27,20,25,17)(13,21,28,33,30,35,29,18)(24,36,39,37,40,34,38,31);(1,4,5,3)(2,6,7,8)(9,12,14,25,16,27,15,26)(10,17,11,20,19,32,22,23)(13,24,28,38,30,40,29,39)(18,31,21,34,33,37,35,36);(2,7)(6,8)(9,13,15,29,16,30,14,28)(10,18,22,35,19,33,11,21)(12,24,26,39,27,40,25,38)(17,31,23,36,32,37,20,34)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=e,d^2=fg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 279728);