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 | |
11942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22);(5,6)(7,8)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24);(2,3)(10,12)(11,17)(16,20)(19,23);(1,3,4,2)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,e^2=i,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 | #72312270, #72318065, #72318091, … |
| maximal normal subgroup of | #72312270, #72318065, #72318091, … |
| maximal quotient of | #72335299, #72335304, #72335648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581687);