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 | |
7938 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)(5,6,9,7)(8,11,12,10)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(6,7)(10,11)(14,15)(16,23)(20,22)(24,28);(5,7,9,6)(8,11,12,10)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237029, #72327640, #72330769, … |
| maximal normal subgroup of | #72237029, #72327640, #72330769, … |
| maximal quotient of | #72327713, #72327963, #72328167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7620397);