Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #392 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C4×#392 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1827 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)(3,14,17,9,19,31,18,32)(4,20,22,10,23,36,28,35)(5,11,25,37,26,38,24,52)(15,33,43,39,44,29,42,48)(16,40,45,30,47,55,46,56)(21,49,50,34,51,59,53,58)(41,57,62,60,63,54,61,64);(1,3)(2,9)(4,15)(5,16)(6,18)(7,17)(8,19)(10,33)(11,30)(12,31)(13,32)(14,27)(20,48)(21,41)(22,43)(23,44)(24,46)(25,45)(26,47)(28,42)(29,35)(34,57)(36,39)(37,55)(38,56)(40,52)(49,64)(50,62)(51,63)(53,61)(54,58)(59,60);(1,4)(2,10)(3,15)(5,21)(6,28)(7,22)(8,23)(9,29)(11,34)(12,36)(13,35)(14,39)(16,41)(17,43)(18,42)(19,44)(20,27)(24,53)(25,50)(26,51)(30,54)(31,48)(32,33)(37,59)(38,58)(40,60)(45,62)(46,61)(47,63)(49,52)(55,64)(56,57);(1,5,7,25,8,26,6,24)(2,11,12,37,13,38,27,52)(3,16,17,45,19,47,18,46)(4,21,22,50,23,51,28,53)(9,30,31,55,32,56,14,40)(10,34,36,59,35,58,20,49)(15,41,43,62,44,63,42,61)(29,54,48,64,33,57,39,60)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285493, C2×#359071, #72285688, … |
| maximal normal subgroup of | #72285493, C2×#359071, #72285688, … |
| maximal quotient of | C2×#359071, #72343059 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266267);