#496182

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21738
prime radical	C₂×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403378);