#232834

order	512
nilpotency class	3
isoclinism	isoclinic class of #232834
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1302 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)(3,10)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3)(2,7)(4,13)(5,11)(6,12)(8,15)(9,16)(10,14)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(3,11)(4,5)(7,15)(8,14)(10,16)(12,13)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30);(3,12)(7,16)(10,15)(11,13)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140030);