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 | |
5925 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)(9,10)(11,16)(12,13)(14,15)(17,18)(19,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,12,13)(17,19,21,20)(18,22,23,24);(2,5)(6,8)(10,15)(11,12)(18,23)(22,24);(9,11,14,12)(10,16,15,13)(17,18)(19,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,15,16);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,19)(18,24,23,22)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305479 |
| maximal normal subgroup of | #72305479 |
| maximal quotient of | #72306263, #72306266 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577786);