Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,18,19)(15,21,24,20)(16,17,25,26)(22,27,28,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(14,19)(16,25)(20,21)(22,28);(13,15,18,24)(14,20,19,21)(16,22,25,28)(17,23,26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,17)(15,22)(18,25)(19,26)(20,23)(21,27)(24,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,16)(15,23)(18,26)(19,25)(20,22)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72310471, #72342348 |
| maximal normal subgroup of | #72310471, #72342348 |
| maximal quotient of | #72310492 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549522);