#409165

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316361);