#500820

order	512
nilpotency class	4
isoclinism	isoclinic class of #500819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24260
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1029 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408016);