#240786

order	512
nilpotency class	3
isoclinism	isoclinic class of #240771
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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)(17,18,23,28)(19,29,33,44)(20,34,37,25)(21,26,22,27)(24,31,40,32)(30,45,47,41)(35,39,46,48)(36,43,38,42);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,23,33)(18,24,28,40)(20,35,37,46)(21,31,22,32)(25,41,34,45)(26,29,27,44)(30,36,47,38)(39,42,48,43);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20,21,36,23,37,22,38)(18,25,27,43,28,34,26,42)(19,30,32,46,33,47,31,35)(24,39,29,41,40,48,44,45);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,29,40,44)(25,42,34,43)(30,35,47,46)(39,41,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=h,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=gh,d^c=di,f^c=fhi

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, 147982);