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 | |
7477 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)(13,14)(15,23)(16,18)(17,20)(19,22)(21,27)(24,28)(25,26);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11);(5,7,9,10)(6,8,11,12)(13,15,19,17)(14,20,22,23)(16,24,27,26)(18,25,21,28);(13,16)(14,21)(15,24)(17,26)(18,22)(19,27)(20,28)(23,25);(5,6)(7,8)(9,11)(10,12)(13,17,19,15)(14,20,22,23)(16,26,27,24)(18,25,21,28);(5,8)(6,10)(7,11)(9,12)(13,18,19,21)(14,16,22,27)(15,25,17,28)(20,24,23,26)Polycyclic presentation:
2^9|c^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238226, #72238230, #72238252 |
| maximal normal subgroup of | #72238226, #72238230, #72238252 |
| maximal quotient of | #72263539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561792);