#217049

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124245);