#240849

order	512
nilpotency class	3
isoclinism	isoclinic class of #240845
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

895 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,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,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,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, 148045);