#438757

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

796 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345953);