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 | |
8136 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)(5,6)(7,11)(8,9)(10,12)(13,14,19,20)(15,18,23,28)(16,21,25,27)(17,26,24,22);(1,2,3,4)(13,15,19,23)(14,18,20,28)(16,24,25,17)(21,22,27,26);(6,9)(10,11)(14,20)(15,23)(16,25)(22,26);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,24)(17,23)(18,22)(19,25)(20,27)(26,28);(5,7,8,12)(6,10,9,11)(13,17,19,24)(14,22,20,26)(15,16,23,25)(18,27,28,21);(13,18)(14,23)(15,20)(16,22)(17,21)(19,28)(24,27)(25,26)Polycyclic presentation:
2^9|a^2=i,b^2=gi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72280398, #72331611 |
| maximal normal subgroup of | #72280398, #72331611 |
| maximal quotient of | #72280735, #72280852, #72281350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7764723);