#237395

order	512
nilpotency class	3
isoclinism	isoclinic class of #237395
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

1508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

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=gi,c^b=cg,e^b=eh,g^b=gi,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

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, 144591);