#254301

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36607	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1050 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)(3,13)(4,9)(5,22)(6,10)(7,12)(8,17)(11,24)(14,16)(15,18)(19,30)(20,29)(21,28)(23,25)(26,32)(27,31);(1,2,4,9)(3,14,16,13)(5,22,6,10)(7,12,21,28)(8,17,26,32)(11,25,23,24)(15,18,27,31)(19,29,20,30);(1,3,4,20)(2,8,9,15)(5,18,6,17)(7,19,21,16)(10,29,22,13)(11,30,23,14)(12,27,28,26)(24,32,25,31);(1,2,4,9)(3,15,20,8)(5,23,6,11)(7,12,21,28)(10,25,22,24)(13,31,29,32)(14,18,30,17)(16,27,19,26)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161497);