#395864

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1646 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)(11,16)(12,14)(13,15)(17,18,20,25,22,26,21,24)(19,23,27,31,29,32,28,30);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,29)(18,23,26,32)(20,27,21,28)(24,30,25,31);(2,4)(6,8)(9,12,13,11)(10,14,15,16)(18,24)(20,21)(23,30)(25,26)(27,28)(31,32);(18,25)(20,21)(23,31)(24,26)(27,28)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi,h^c=hi,f^d=fi,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, 303060);