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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,9)(7,11)(13,14)(15,22)(16,17)(18,19)(20,23)(21,26)(24,28)(25,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(15,23)(20,22)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(14,19)(16,21)(20,22)(24,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,23)(14,20,19,22)(16,24,21,27)(17,25,26,28);(13,16,18,21)(14,17,19,26)(15,24,23,27)(20,25,22,28);(13,17)(14,21)(15,25)(16,19)(18,26)(20,27)(22,24)(23,28)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=hi,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 | #72256638, #72256694, #72256700, … |
| maximal normal subgroup of | #72256638, #72256694, #72256700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551541);