Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187949 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #62817 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,7)(4,8)(5,10)(6,11)(9,18)(13,26,27,31)(14,20,29,21)(16,24)(17,22)(19,28,25,32)(23,30);(1,3)(2,7)(4,13)(5,15)(6,14)(8,19)(9,20)(10,12)(11,21)(16,27)(17,28)(18,29)(22,31)(23,26)(24,25)(30,32);(1,4)(2,8)(3,13)(5,16)(6,17)(7,19)(9,23)(10,24)(11,22)(12,25)(14,28)(15,27)(18,30)(20,26)(21,31)(29,32);(2,9)(4,16)(7,20)(8,22)(10,11)(12,21)(13,27)(17,30)(19,31)(23,24)(25,26)(28,32)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72211144, #72211186, #72225907, … |
| maximal normal subgroup of | #72211144, #72211186, #72225907, … |
| maximal quotient of | #72225624, #72225627, #72225675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95145);