#237111

order	512
nilpotency class	3
isoclinism	isoclinic class of #237079
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1239 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,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, 144307);