Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141878 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1988 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11384 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,14,18,45)(4,19,22,54)(5,24)(6,12)(7,30)(8,13)(9,16,34,51)(10,21,38,23)(11,28)(15,46)(17,44,31,39)(20,40,26,37)(25,35,61,33)(27,41)(29,42)(32,60)(36,55)(43,58)(47,63)(48,62)(49,64)(50,52)(53,56,59,57);(1,3)(2,9,13,34)(4,15)(5,25)(6,17)(7,31)(8,18)(10,36,38,32)(11,39,41,44)(12,33,30,35)(14,42,45,24)(16,28)(19,52,54,63)(20,55)(21,48)(22,49)(23,58)(26,60)(27,51)(29,61)(37,64,40,46)(43,59,62,53)(47,57)(50,56);(1,4)(2,10,13,38)(3,15)(5,26)(6,21)(7,23)(8,22)(9,32,34,36)(11,40,41,37)(12,19,30,54)(14,43,45,62)(16,50)(17,48)(18,49)(20,29)(24,59,42,53)(25,60)(27,57)(28,56)(31,58)(33,63,35,52)(39,64,44,46)(47,51)(55,61)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,g^b=gi,h^b=hi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279896, #72280652, #72280708, … |
| maximal normal subgroup of | #72279896, #72280652, #72280708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56658);