#372534

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

1217 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)(9,10,15,19,16,21,14,20)(11,12,24,27,25,28,23,17)(13,29,31,35,32,34,30,18)(22,36,38,40,39,33,37,26);(1,3,5,4)(2,6,7,8)(9,11)(10,17,21,27)(12,20,28,19)(13,22)(14,23)(15,24)(16,25)(18,33,35,36)(26,34,40,29)(30,37)(31,38)(32,39);(1,4,5,3)(2,6,7,8)(9,12,14,27,16,28,15,17)(10,11,19,23,21,25,20,24)(13,26,30,36,32,40,31,33)(18,22,29,37,35,39,34,38);(2,7)(6,8)(9,13,15,31,16,32,14,30)(10,18,20,34,21,35,19,29)(11,22,24,38,25,39,23,37)(12,26,17,33,28,40,27,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=fgi,e^2=i,f^2=i,h^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, 279730);