#459760

order	512
nilpotency class	3
isoclinism	isoclinic class of #459760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16348
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366956);