#240724

order	512
nilpotency class	3
isoclinism	isoclinic class of #240721
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	C₂×#532	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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,d^c=di,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, 147920);