#391783

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298979);