#364123

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1742 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=i,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=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, 271319);