Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14127 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,17,9,19,32,18,33)(4,20,22,10,23,37,28,36)(5,24,25,11,26,39,29,38)(15,34,44,40,45,30,43,49)(16,31,47,55,48,56,46,41)(21,35,52,58,53,59,51,50)(42,61,62,57,64,60,63,54);(1,3)(2,9)(4,15)(5,16)(6,18)(7,17)(8,19)(10,34)(11,31)(12,32)(13,33)(14,27)(20,49)(21,42)(22,44)(23,45)(24,41)(25,47)(26,48)(28,43)(29,46)(30,36)(35,57)(37,40)(38,56)(39,55)(50,61)(51,63)(52,62)(53,64)(54,59)(58,60);(1,4,7,22,8,23,6,28)(2,10,12,37,13,36,27,20)(3,15,17,44,19,45,18,43)(5,21,25,52,26,53,29,51)(9,30,32,49,33,34,14,40)(11,35,39,58,38,59,24,50)(16,42,47,62,48,64,46,63)(31,54,55,61,56,57,41,60);(1,5)(2,11)(3,16)(4,21)(6,29)(7,25)(8,26)(9,31)(10,35)(12,39)(13,38)(14,41)(15,42)(17,47)(18,46)(19,48)(20,50)(22,52)(23,53)(24,27)(28,51)(30,54)(32,55)(33,56)(34,57)(36,59)(37,58)(40,60)(43,63)(44,62)(45,64)(49,61)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285688, #72285784, #72286087, … |
| maximal normal subgroup of | #72285688, #72285784, #72286087, … |
| maximal quotient of | C2×#373685, #72343142 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280881);