Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
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 | |
6120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,8,4,7)(10,13)(14,16)(17,18,21,23)(19,20,24,22);(1,3,6,4)(2,7,5,8)(10,13)(11,15)(17,19,21,24)(18,22,23,20);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,22)(23,24);(1,4,6,3)(2,7,5,8);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72213471 |
| maximal normal subgroup of | #72213471 |
| maximal quotient of | #72213467, #72266996, #72266999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004960);