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 | |
7902 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)(5,6,9,10)(7,8,11,12)(13,14)(15,18)(16,20)(17,21)(19,22)(23,25)(24,27)(26,28);(1,2,3,4)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(6,10)(7,11)(15,22)(16,24)(18,19)(20,27);(5,7)(6,8)(9,11)(10,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27);(5,8)(6,7)(9,12)(10,11)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28);(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72323293, C2×#7739461 |
| maximal normal subgroup of | #72323293, C2×#7739461 |
| maximal quotient of | #72323296, #72323361, #72323386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646657);