#240660

order	512
nilpotency class	3
isoclinism	isoclinic class of #240653
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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)(3,11)(4,7)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,31)(25,36)(29,46)(30,40)(32,44)(34,45)(35,43)(37,42)(39,48)(41,47);(2,7)(3,12)(4,5)(8,11)(9,10)(15,16)(17,19,23,32)(18,24,27,40)(20,34,36,29)(21,31,22,30)(25,41,33,39)(26,44,38,28)(35,47,37,48)(42,46,43,45);(1,3,4,13,6,12,5,14)(2,8,9,15,10,16,7,11)(17,20,22,35,23,36,21,37)(18,25,26,43,27,33,38,42)(19,29,31,47,32,34,30,48)(24,39,28,45,40,41,44,46);(1,4,6,5)(2,7,10,9)(3,13,12,14)(8,11,16,15)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,35,36,37)(24,28,40,44)(25,42,33,43)(29,47,34,48)(39,46,41,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=e,d^2=h,e^2=h,f^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,f^b=fi,d^c=dhi,f^c=fhi,g^c=gh

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, 147856);