#410791

order	512
nilpotency class	3
isoclinism	isoclinic class of #403107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2547
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15090
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

653 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)(3,13,22,24,15,8,16,25)(4,17,21,28,19,9,20,27)(14,23,18,31,30,29,32,26);(1,3,6,16,7,15,5,22)(2,8,12,25,11,13,10,24)(4,18,21,30,19,32,20,14)(9,26,28,29,17,31,27,23);(1,4,5,20,7,19,6,21)(2,9,12,28,11,17,10,27)(3,14,16,18,15,30,22,32)(8,23,24,31,13,29,25,26);(2,10,11,12)(8,24,13,25)(9,27,17,28)(23,31,29,26)

Polycyclic presentation:

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

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, 317987);