Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(2,5)(6,8)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32);(10,13)(14,16)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(9,11,12,15)(10,14,13,16)(19,26)(20,28)(22,30)(23,32)(25,27)(29,31);(1,3,4,7)(2,6,5,8)(19,27)(22,31)(25,26)(29,30)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257326, #72257411, #72257418, … |
| maximal normal subgroup of | #72257326, #72257411, #72257418, … |
| maximal quotient of | #72264301, #72308429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356841);