Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493477 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62862 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3484 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2309 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)(5,17,18,9)(6,10)(7,14,19,26)(12,23)(13,24)(15,28,29,22)(16,21)(20,27,30,32)(25,31);(1,3)(2,7)(4,13)(5,14)(6,12)(8,20)(9,11)(10,19)(15,27)(16,25)(17,23)(18,26)(21,30)(22,24)(28,31)(29,32);(3,12)(7,19)(8,21)(11,23)(13,25)(14,26)(22,28)(27,32);(1,4)(2,8)(3,13)(5,15)(6,16)(7,20)(9,22)(10,21)(11,24)(12,25)(14,27)(17,28)(18,29)(19,30)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,g^b=gh,d^c=dh,f^d=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72232878, #72233206, #72249650, … |
| maximal normal subgroup of | #72232878, #72233206, #72249650, … |
| maximal quotient of | #72232778, #72233184, #72237484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400673);