Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
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 | |
8802 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)(17,25)(22,26)(23,28)(24,27);(1,2)(3,4)(6,9)(10,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,27)(24,28)(25,26);(5,6,8,9)(7,10,11,12)(14,19)(16,20)(22,26)(24,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,21)(14,20,19,16)(17,23,25,28)(22,27,26,24);(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23);(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72327246, #72342297, #72342346 |
| maximal normal subgroup of | #72327246, #72342297, #72342346 |
| maximal quotient of | #72327361, #72327431, #72327976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550414);