#237259

order	512
nilpotency class	3
isoclinism	isoclinic class of #237257
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×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|b^2=h,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,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,g^c=gi

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, 144455);