Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493483 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62876 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3484 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2308 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)(3,11,12,24)(4,8)(5,17,19,9)(6,10)(7,14)(13,25,26,32)(15,28,30,23)(16,22)(18,20)(21,27)(29,31);(1,3)(2,7)(4,13)(5,18)(6,12)(8,21)(9,24)(10,20)(11,17)(14,19)(15,29)(16,26)(22,31)(23,32)(25,28)(27,30);(3,12)(7,20)(8,22)(11,24)(13,26)(14,18)(23,28)(27,29);(1,4)(2,8)(3,13)(5,15)(6,16)(7,21)(9,23)(10,22)(11,25)(12,26)(14,27)(17,28)(18,29)(19,30)(20,31)(24,32)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72233206, #72249653, #72258019, … |
| maximal normal subgroup of | #72233206, #72249653, #72258019, … |
| maximal quotient of | #72233190, #72234416, #72238631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400679);