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 | |
6229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,24);(1,2)(3,8)(4,7)(5,6)(17,18)(19,23)(20,24)(21,22);(2,6)(3,7)(10,13)(14,16);(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,22,24);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23);(1,4)(2,7)(3,6)(5,8)Polycyclic presentation:
2^9|d^2=gi,e^2=gh,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 | #72232449, #72263266 |
| maximal normal subgroup of | #72232449, #72263266 |
| maximal quotient of | #72232476, #72232500, #72263310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7583565);