#232869

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

1182 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 140065);