Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(10,12)(14,15)(18,22)(20,23);(2,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18)(19,20)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,22)(19,23,24,20);(2,5)(6,8)(11,16)(14,15)(17,19,21,24)(18,20,22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72233593, #72233609, #72260820 |
| maximal normal subgroup of | #72233593, #72233609, #72260820 |
| maximal quotient of | #72262092 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538205);