#240588

order	512
nilpotency class	3
isoclinism	isoclinic class of #240579
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

790 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)(4,6)(5,9)(7,10)(8,13)(12,15)(14,16)(17,18)(19,28)(20,32)(21,26)(22,37)(23,27)(24,30)(25,36)(29,45)(31,40)(33,44)(34,47)(35,43)(38,41)(39,48)(42,46);(1,3,7,15)(2,8,10,16)(4,11,9,12)(5,13,6,14)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,42)(26,40)(27,41)(28,37)(29,36)(32,39)(34,48)(35,46)(43,44)(45,47);(1,4,6,2,7,9,5,10)(3,12,13,8,15,11,14,16)(17,20,22,35,23,36,21,34)(18,25,26,43,27,32,37,47)(19,29,30,46,31,33,38,48)(24,39,28,44,41,42,40,45);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,12,16,11)(17,21,23,22)(18,26,27,37)(19,30,31,38)(20,34,36,35)(24,40,41,28)(25,43,32,47)(29,46,33,48)(39,45,42,44)

Polycyclic presentation:

2^9|b^2=i,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,e^b=eh,f^b=fi,d^c=di,f^c=fhi,g^c=gh

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, 147784);