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 | |
12106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24);(5,6,7,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24);(1,2)(3,4)(11,12)(15,18)(19,20)(23,24);(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)Polycyclic presentation:
2^9|b^2=g,d^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72318009, #72318091, #72318259, … |
| maximal normal subgroup of | #72318009, #72318091, #72318259, … |
| maximal quotient of | #72327602, #72327724, #72327867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537699);