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 | |
5984 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)(3,8)(4,7)(5,6)(18,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,18,21,22)(19,23,20,24);(2,6)(3,7)(10,15)(11,12);(9,11,14,12)(10,16,15,13)(17,19,21,20)(18,23,22,24);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,13,15,16);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|b^2=g,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72262868, #72262935, #72262962 |
| maximal normal subgroup of | #72262868, #72262935, #72262962 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644620);