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 | |
8090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,24,19)(16,26,27,20)(17,21,28,25);(1,2,3,4)(5,7,9,8)(6,10,12,11)(15,24)(17,28)(19,23)(21,25);(7,8)(10,11)(13,15,18,24)(14,19,22,23)(16,21,27,25)(17,20,28,26);(5,7,9,8)(6,11,12,10);(5,8,9,7)(6,10,12,11)(13,16,18,27)(14,20,22,26)(15,25,24,21)(17,23,28,19);(5,6)(7,11)(8,10)(9,12)(13,17)(14,21)(15,20)(16,23)(18,28)(19,27)(22,25)(24,26)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304562, #72304592, #72304597, … |
| maximal normal subgroup of | #72304562, #72304592, #72304597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644709);