Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13977 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,19,32,18,39,17,9)(4,10,24,34,23,35,22,53)(5,25,27,36,28,54,26,11)(15,40,43,30,20,37,42,56)(16,44,47,59,46,31,45,38)(21,49,52,57,51,64,50,33)(41,55,62,58,48,63,61,60);(1,3)(2,9,13,32)(4,20)(5,16)(6,17)(7,19)(8,18)(10,30,35,56)(11,31,36,44)(12,14,29,39)(15,23)(21,48)(22,43)(24,42)(25,38,54,59)(26,45)(27,47)(28,46)(33,55,57,63)(34,37,53,40)(41,51)(49,58,64,60)(50,62)(52,61);(1,4)(2,10)(3,15,18,20)(5,21)(6,22)(7,24)(8,23)(9,30,32,56)(11,33)(12,34)(13,35)(14,37,39,40)(16,41,46,48)(17,42,19,43)(25,49)(26,50)(27,52)(28,51)(29,53)(31,55,44,63)(36,57)(38,58,59,60)(45,61,47,62)(54,64);(1,5,6,26,8,28,7,27)(2,11,12,25,13,36,29,54)(3,16,17,45,18,46,19,47)(4,21,22,50,23,51,24,52)(9,31,14,38,32,44,39,59)(10,33,34,49,35,57,53,64)(15,41,42,61,20,48,43,62)(30,55,37,58,56,63,40,60)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319834, #72319846, #72319889, … |
| maximal normal subgroup of | #72319834, #72319846, #72319889, … |
| maximal quotient of | #72343146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272826);