Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9000 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)(6,9)(7,11)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28);(1,2)(3,4)(6,9)(10,12)(15,22)(19,21)(23,28)(26,27);(5,6)(7,10)(8,9)(11,12)(14,18)(19,21)(20,24)(26,27);(5,7,8,11)(6,10,9,12)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27);(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72331266 |
| maximal normal subgroup of | #72331266 |
| maximal quotient of | #72331294, #72332154, #72332372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550815);