Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,22,20,24);(2,5)(6,8)(11,15)(13,16)(17,19,21,20)(18,22,23,24);(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,23);(10,14)(11,15)(17,20,21,19)(18,22,23,24);(11,15)(13,16)(18,23)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72303876, #72312021 |
| maximal normal subgroup of | #72303876, #72312021 |
| maximal quotient of | #72304035, #72304053 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609285);