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 | |
8976 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)(7,11)(10,12)(13,14,19,21)(15,22,17,20)(16,18,25,28)(23,27,26,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,17)(14,20,21,22)(16,23,25,26)(18,24,28,27);(6,9)(10,12)(14,21)(15,17)(16,25)(24,27);(13,16)(14,18)(15,23)(17,26)(19,25)(20,24)(21,28)(22,27);(5,6)(7,10)(8,9)(11,12)(13,17,19,15)(14,22,21,20)(16,26,25,23)(18,27,28,24);(5,7)(6,10)(8,11)(9,12)(13,18)(14,16)(15,24)(17,27)(19,28)(20,23)(21,25)(22,26)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304538, #72304569, #72304580, … |
| maximal normal subgroup of | #72304538, #72304569, #72304580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615816);