Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511793 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2476 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,24)(14,25)(15,21)(16,22)(17,30)(18,23)(19,27)(20,29)(26,32)(28,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,26)(16,27)(17,28)(18,29)(21,31)(22,24)(23,25)(30,32);(1,4)(2,8)(3,12)(5,17)(6,15)(7,19)(9,22)(10,21)(11,24)(13,28)(14,26)(16,18)(20,31)(23,30)(25,32)(27,29);(4,15)(5,18)(8,21)(9,23)(11,25)(12,26)(13,29)(19,31)Polycyclic presentation:
2^9|b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233205, #72233206, #72235130, … |
| maximal normal subgroup of | #72233205, #72233206, #72235130, … |
| maximal quotient of | #72213329, #72233086, #72233141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418989);