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 | |
11339 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)(9,10,15,12)(11,16,18,13)(14,20,22,17)(19,21,24,23);(5,6)(7,8)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24);(2,4)(11,18)(13,16)(19,24)(21,23);(1,2)(3,4)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23);(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23);(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^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 | #72234472, #72234475, #72234480, … |
| maximal normal subgroup of | #72234472, #72234475, #72234480, … |
| maximal quotient of | #72261681, #72261699, #72261717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7588957);