Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134702 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1809 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11231 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
465 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)(3,15,18,39)(4,20,23,36)(5,25)(6,28)(7,31)(8,13)(9,17,33,29)(10,22,37,21)(11,27)(12,30)(14,26)(16,42)(19,48,47,41)(24,55,51,50)(32,44)(34,56,57,46)(35,58)(38,52,49,54)(40,53)(43,59)(45,60)(61,62)(63,64);(1,3,5,17,7,19,27,46,8,18,6,29,26,47,30,56)(2,9,11,15,14,34,25,41,13,33,12,39,31,57,28,48)(4,16,21,43,24,44,52,62,23,45,22,53,51,58,54,64)(10,35,36,59,38,60,55,63,37,32,20,40,49,42,50,61);(1,4,5,21,7,24,27,52,8,23,6,22,26,51,30,54)(2,10,11,36,14,38,25,55,13,37,12,20,31,49,28,50)(3,16,17,43,19,44,46,62,18,45,29,53,47,58,56,64)(9,32,15,40,34,42,41,61,33,35,39,59,57,60,48,63)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307185, #72307209, #72307292, … |
| maximal normal subgroup of | #72307185, #72307209, #72307292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56072);