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 | |
6150 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)(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(18,21)(22,24);(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24);(1,4,5,6)(2,3,7,8)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^9|e^2=ghi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233690, #72233694, #72304331, … |
| maximal normal subgroup of | #72233690, #72233694, #72304331, … |
| maximal quotient of | #72303651, #72304040 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591311);