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 | |
7762 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)(13,14)(15,23)(16,25)(17,22)(18,20)(19,24)(21,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,18,24)(14,19,20,23)(16,22,26,28)(17,25,27,21);(5,7)(6,9)(8,12)(10,11)(14,20)(15,24)(21,25)(22,28);(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,22)(17,19)(18,26)(20,25)(23,27)(24,28);(5,6)(7,9)(8,11)(10,12)(13,17)(14,22)(15,25)(16,23)(18,27)(19,26)(20,28)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^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 | #72331666, #72332008, #72332087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7757062);