#232865

order	512
nilpotency class	3
isoclinism	isoclinic class of #232862
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

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(3,10)(4,5)(7,11)(8,14)(12,13)(15,16)(17,18)(19,28)(20,25)(21,36)(22,27)(23,26)(24,30)(29,45)(31,39)(32,44)(33,38)(34,43)(35,41)(37,42)(40,46)(47,48);(1,2)(3,11)(4,8)(5,14)(6,9)(7,13)(10,15)(12,16)(17,19,23,32)(18,24,26,39)(20,33,35,47)(21,31,22,30)(25,40,41,45)(27,44,36,28)(29,37,46,34)(38,42,48,43);(1,3)(2,7)(4,13)(5,10)(6,12)(8,16)(9,15)(11,14)(17,20,23,35)(18,25,26,41)(19,29,32,46)(21,34,22,37)(24,38,39,48)(27,42,36,43)(28,45,44,40)(30,33,31,47);(18,26)(24,39)(25,41)(27,36)(28,44)(38,48)(40,45)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140061);