#240664

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,27,29,42)(20,31,33,24)(21,35,37,25)(23,36,39,30)(28,44,32,43)(34,41,48,46)(38,45,40,47);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,22,29)(18,23,26,39)(20,32,33,28)(21,36,37,30)(24,40,31,38)(25,42,35,27)(34,45,48,47)(41,43,46,44);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20,21,34,22,33,37,48)(18,24,25,41,26,31,35,46)(19,28,30,45,29,32,36,47)(23,38,27,43,39,40,42,44);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(19,29)(20,33)(23,39)(24,31)(27,42)(30,36)(34,48)(41,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,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, 147860);