#496471

order	512
nilpotency class	3
isoclinism	isoclinic class of #360200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2311
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21851
prime radical	#3444	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

757 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=ce,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403667);