#237365

order	512
nilpotency class	3
isoclinism	isoclinic class of #237345
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

1200 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)(4,7)(5,9)(6,15)(8,13)(11,12)(14,16)(18,23)(19,26)(20,21)(25,30)(27,31)(28,29);(1,3,7,14)(2,8,4,11)(5,13,6,12)(9,16,15,10)(17,18,22,25)(19,27,29,31)(20,30,21,23)(24,26,32,28);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,19,22,29)(18,24,25,32)(20,26,21,28)(23,31,30,27);(1,5,7,6)(2,9,4,15)(3,12,14,13)(8,10,11,16)(18,25)(23,30)(24,32)(27,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,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=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, 144561);