Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417496 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,20)(21,26)(22,25)(23,27)(24,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,21)(17,22)(19,26)(20,25)(23,28)(24,27);(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,28)(25,26);(2,4)(7,12)(9,11)(16,23)(19,27)(21,28)(24,26)Polycyclic presentation:
2^9|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285059, #72285073, #72285094, … |
| maximal normal subgroup of | #72285059, #72285073, #72285094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325089);