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 | |
7714 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)(15,23)(16,21)(17,26)(18,20)(19,24)(22,27)(25,28);(1,2,3,4)(13,15)(14,19)(16,17)(18,24)(20,23)(21,22)(25,27)(26,28);(6,9)(10,11)(14,20)(15,24)(16,25)(22,26);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,17)(18,25)(19,22)(20,28)(23,26)(24,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,25)(16,24)(18,27)(19,28)(20,26)(21,23);(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,28)(26,27)Polycyclic presentation:
2^9|b^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72330752 |
| maximal normal subgroup of | #72330752 |
| maximal quotient of | #72330935, #72331596, #72331720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7694459);