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 | |
8530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6,9,10)(7,8,11,12)(13,14,19,16)(15,17,21,22)(18,20,26,24)(23,25,27,28);(1,2)(3,4)(13,15)(14,17)(16,22)(18,23)(19,21)(20,25)(24,28)(26,27);(6,10)(7,11)(14,16)(15,21)(18,26)(25,28);(5,7)(6,8)(9,11)(10,12)(13,16,19,14)(15,22,21,17)(18,24,26,20)(23,28,27,25);(5,8)(6,7)(9,12)(10,11)(13,17,19,22)(14,15,16,21)(18,25,26,28)(20,23,24,27);(13,18)(14,20)(15,23)(16,24)(17,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=gh,d^2=i,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256659, #72256663, #72256671, … |
| maximal normal subgroup of | #72256659, #72256663, #72256671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7703006);