Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2280 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14681 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3387 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)(3,14,18,8)(4,19,21,27)(5,12)(6,13)(7,11)(9,16,24,17)(10,20,26,22)(15,29)(23,30)(25,31)(28,32);(1,3)(2,8)(4,15)(5,16)(6,17)(7,18)(9,12)(10,23)(11,14)(13,24)(19,25)(20,30)(21,31)(22,32)(26,28)(27,29);(1,3)(2,9,11,24)(4,15)(5,16)(6,17)(7,18)(8,12,14,13)(10,25,26,29)(19,28,27,23)(20,30)(21,31)(22,32);(1,4)(2,10)(3,15)(5,20)(6,22)(7,21)(8,23)(9,25)(11,26)(12,19)(13,27)(14,28)(16,30)(17,32)(18,31)(24,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286033, #72286134, #72286267, … |
| maximal normal subgroup of | #72286033, #72286134, #72286267, … |
| maximal quotient of | C2×#392276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299472);