#240967

order	512
nilpotency class	3
isoclinism	isoclinic class of #240963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,d^c=di,f^c=fhi,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, 148163);