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 | |
8458 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)(6,10)(7,12)(13,14)(15,22)(16,20)(17,25)(18,19)(21,26)(23,24)(27,28);(5,6,9,10)(7,11,12,8)(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,28)(26,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(14,19)(15,20)(21,25)(23,27);(13,14)(15,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,28);(13,16)(14,20)(15,19)(17,24)(18,22)(21,27)(23,25)(26,28);(5,8,9,11)(6,7,10,12)(13,17,18,26)(14,21,19,25)(15,23,20,27)(16,24,22,28)Polycyclic presentation:
2^9|b^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72304534, #72304584, #72304616, … |
| maximal normal subgroup of | #72304534, #72304584, #72304616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538241);