#232637

order	512
nilpotency class	3
isoclinism	isoclinic class of #232619
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	#60949	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1232 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:

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

Polycyclic presentation:

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

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, 139833);