#387385

order	512
nilpotency class	3
isoclinism	isoclinic class of #384166
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2377
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14470
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=efg,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=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, 294581);