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 | |
11312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,22)(23,24);(5,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,22)(23,24);(2,4)(10,15)(11,19)(17,20)(18,24);(1,2)(3,4)(9,11,14,19)(10,16,15,21)(12,18,22,24)(13,20,23,17);(9,12)(10,17)(11,18)(13,16)(14,22)(15,20)(19,24)(21,23);(9,13)(10,18)(11,20)(12,21)(14,23)(15,24)(16,22)(17,19)Polycyclic presentation:
2^9|d^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72234477, #72234485, #72256674, … |
| maximal normal subgroup of | #72234477, #72234485, #72256674, … |
| maximal quotient of | #72261363, #72261371, #72261384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592050);