#409368

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15018
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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:

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

Polycyclic presentation:

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

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, 316564);