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 | |
7816 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,18)(16,20)(17,21)(19,22)(23,25)(24,27)(26,28);(1,2,3,4)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(6,9)(10,11)(15,22)(16,24)(18,19)(20,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27);(5,7)(6,10)(8,12)(9,11)(13,16,17,24)(14,20,21,27)(15,23,22,28)(18,25,19,26);(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|b^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72330769 |
| maximal normal subgroup of | #72330769 |
| maximal quotient of | #72330860, #72331339, #72332067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7649621);