#237321

order	512
nilpotency class	3
isoclinism	isoclinic class of #237305
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

1300 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(18,23)(19,26)(20,21)(25,30)(27,31)(28,29);(2,7)(4,5)(8,14)(9,10)(11,13)(15,16)(17,18)(19,27)(20,30)(21,23)(22,25)(24,29)(26,32)(28,31);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,15)(10,16)(17,19)(18,24)(20,26)(21,29)(22,28)(23,31)(25,32)(27,30);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,14,16,15)(18,25)(19,28)(23,30)(26,29)

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=dh,e^c=ei,f^c=fi,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, 144517);