#383290

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290486);