#403089

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14867
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310285);