Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
12358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(5,6,7,8)(9,10,13,14)(11,15,17,21)(12,16,20,19)(18,22,23,24);(2,4)(10,14)(11,17)(16,19)(18,23);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)Polycyclic presentation:
2^9|b^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72234473, #72234477, #72234484, … |
| maximal normal subgroup of | #72234473, #72234477, #72234484, … |
| maximal quotient of | #72240163, #72252053, #72252054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579971);