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 | |
11123 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)(5,6,7,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,23)(21,24);(5,6)(7,8)(9,11,14,21)(10,15,18,24)(12,19,22,17)(13,20,23,16);(2,4)(9,12,14,22)(10,16,18,20)(11,17,21,19)(13,15,23,24);(1,2)(3,4)(9,12)(10,16)(11,19)(13,15)(14,22)(17,21)(18,20)(23,24);(9,10)(11,15)(12,16)(13,19)(14,18)(17,23)(20,22)(21,24);(9,13)(10,17)(11,20)(12,15)(14,23)(16,21)(18,19)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234472, #72234477, #72234480 |
| maximal normal subgroup of | #72234472, #72234477, #72234480 |
| maximal quotient of | #72261727, #72261744, #72261765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604700);