Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C22⋊C4×C4⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,18,32,19,40,29,9)(4,20,23,34,24,10,22,35)(5,11,26,37,28,38,27,39)(16,42,45,48,46,41,44,30)(17,36,25,54,47,60,55,31)(21,49,52,58,53,33,51,59)(43,61,62,64,63,57,50,56);(1,3,7,18,8,19,6,29)(2,9,14,15,13,32,12,40)(4,16,23,44,24,46,22,45)(5,25,27,47,28,55,26,17)(10,30,35,41,20,48,34,42)(11,36,39,54,38,60,37,31)(21,50,52,63,53,62,51,43)(33,57,59,64,49,61,58,56);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,30)(11,33)(12,34)(13,20)(14,35)(15,41)(17,43)(18,44)(19,46)(25,50)(26,51)(27,52)(28,53)(29,45)(31,56)(32,48)(36,57)(37,58)(38,49)(39,59)(40,42)(47,63)(54,64)(55,62)(60,61);(1,5,6,26,8,28,7,27)(2,11,12,37,13,38,14,39)(3,17,18,25,19,47,29,55)(4,21,22,51,24,53,23,52)(9,31,15,36,32,54,40,60)(10,33,34,58,20,49,35,59)(16,43,44,50,46,63,45,62)(30,56,41,57,48,64,42,61)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319733, #72319742, #72319762, … |
| maximal normal subgroup of | #72319733, #72319742, #72319762, … |
| maximal quotient of | #72342995 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263431);