#405041

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=efi,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312237);