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 | |
9698 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)(6,9)(7,11)(13,14)(15,21)(16,24)(17,19)(18,23)(20,25)(22,26)(27,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,17,23)(14,18,19,21)(16,22,25,28)(20,26,24,27);(5,6)(7,10)(8,9)(11,12)(14,19)(18,21)(20,24)(26,27);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,22)(17,23)(19,21)(20,26)(24,27)(25,28);(13,14)(15,18)(16,20)(17,19)(21,23)(22,26)(24,25)(27,28);(13,16)(14,20)(15,22)(17,25)(18,26)(19,24)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250106, #72256638, #72256696, … |
| maximal normal subgroup of | #72250106, #72256638, #72256696, … |
| maximal quotient of | #72250108, #72250250, #72297221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550776);