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 | |
11988 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)(13,21)(17,22)(18,23)(20,24);(5,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24);(1,2,3,4)(10,15)(12,19)(17,22)(20,24);(1,2)(3,4)(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)Polycyclic presentation:
2^9|c^2=h,d^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72241070, #72265848, #72265962, … |
| maximal normal subgroup of | #72241070, #72265848, #72265962, … |
| maximal quotient of | #72241117, #72241119, #72280871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7583335);