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 | |
7782 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)(13,14,18,20)(15,19,23,26)(16,24,25,21)(17,22,27,28);(1,2)(3,4)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28);(6,9)(10,11)(14,20)(15,23)(21,24)(22,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28);(5,7,8,12)(6,10,9,11)(13,16,18,25)(14,21,20,24)(15,22,23,28)(17,26,27,19);(13,17)(14,22)(15,21)(16,19)(18,27)(20,28)(23,24)(25,26)Polycyclic presentation:
2^9|a^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72331611 |
| maximal normal subgroup of | #72331611 |
| maximal quotient of | #72331631, #72331959, #72332064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759977);