#449422

order	512
nilpotency class	3
isoclinism	isoclinic class of #449421
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15943
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 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)(3,6)(4,9)(5,19,21,10)(7,11)(8,12)(13,16)(14,20,26,22)(15,23,30,32)(17,24)(18,25)(27,31,28,29);(1,3,7,12,8,11,6,2)(4,9,17,13,18,25,16,24)(5,10,20,26,21,19,22,14)(15,28,29,32,30,27,31,23);(1,4,7,17,8,18,6,16)(2,9,11,24,12,25,3,13)(5,15,22,31,21,30,20,29)(10,23,26,28,19,32,14,27);(1,5)(2,10,12,19)(3,14,11,26)(4,15)(6,20)(7,22)(8,21)(9,23,25,32)(13,27,24,28)(16,29)(17,31)(18,30)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356618);