#240780

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 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)(3,9)(4,14)(5,10)(6,15)(7,11)(8,12)(13,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,45)(31,40)(33,44)(35,39)(36,43)(38,42)(41,47)(46,48);(1,3,7,13)(2,8,11,14)(4,5,12,6)(9,10,16,15)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,41)(26,29)(27,44)(28,40)(30,38)(34,45)(36,47)(37,46)(39,43)(42,48);(1,4,6,3,7,12,5,13)(2,9,10,14,11,16,15,8)(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);(1,5,7,6)(2,10,11,15)(3,12,13,4)(8,16,14,9)(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|b^2=i,c^2=e,d^2=h,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, 147976);