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 | |
8362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6,8,10)(7,9,12,11)(13,14,19,17)(15,23,24,20)(16,21,25,26)(18,27,28,22);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,18)(17,23)(19,24)(21,22)(25,28)(26,27);(2,3)(5,6)(7,11)(8,10)(9,12)(14,17)(15,24)(16,25)(22,27);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,18)(17,26)(19,25)(20,22)(23,27)(24,28);(13,17,19,14)(15,23,24,20)(16,26,25,21)(18,27,28,22);(13,18)(14,22)(15,16)(17,27)(19,28)(20,21)(23,26)(24,25)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72256663, #72256695, #72256697, … |
| maximal normal subgroup of | #72256663, #72256695, #72256697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578776);