#232667

order	512
nilpotency class	3
isoclinism	isoclinic class of #232654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139863);