Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(15,19)(17,20)(21,22,24,26)(23,27,28,25);(2,4)(5,6,8,7)(10,12)(15,19)(17,20)(23,28)(25,27);(9,10)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=efh,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72329979, #72330359, #72330600 |
| maximal normal subgroup of | #72329979, #72330359, #72330600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 64662);