Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395737 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14726 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,18,20,25)(5,11)(6,9)(7,12)(8,16)(10,19,26,21)(14,23)(15,22,28,32)(17,24)(27,29,31,30);(1,3,5,16,7,14,6,17)(2,8,9,13,12,24,11,23)(4,15,19,29,20,28,21,30)(10,22,25,27,26,32,18,31);(2,9)(3,14)(5,6)(8,13)(10,25)(11,12)(15,28)(18,26)(19,21)(22,27)(23,24)(31,32);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,22)(9,25)(11,18)(12,26)(13,27)(14,28)(16,29)(17,30)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302933);