Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14015 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,23,34)(3,8,15,27,16,26,40,51)(4,17,20,42,19,9,28,31)(5,10,22,33,21,32,45,56)(13,29,37,54,36,24,46,48)(14,25,39,50,38,49,59,63)(18,30,44,55,43,41,52,60)(35,47,58,62,57,53,61,64);(1,3,6,15,7,16,23,40)(2,8,12,27,11,26,34,51)(4,13,20,37,19,36,28,46)(5,14,22,39,21,38,45,59)(9,29,31,54,17,24,42,48)(10,25,33,50,32,49,56,63)(18,35,44,58,43,57,52,61)(30,53,55,64,41,47,60,62);(1,4)(2,9,11,17)(3,13)(5,18)(6,20)(7,19)(8,24,26,29)(10,30,32,41)(12,31,34,42)(14,35)(15,37)(16,36)(21,43)(22,44)(23,28)(25,47,49,53)(27,48,51,54)(33,55,56,60)(38,57)(39,58)(40,46)(45,52)(50,62,63,64)(59,61);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,25)(9,30)(11,32)(12,33)(13,35)(15,39)(16,38)(17,41)(19,43)(20,44)(23,45)(24,47)(26,49)(27,50)(28,52)(29,53)(31,55)(34,56)(36,57)(37,58)(40,59)(42,60)(46,61)(48,62)(51,63)(54,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317343, #72317346, #72317348, … |
| maximal normal subgroup of | #72317343, #72317346, #72317348, … |
| maximal quotient of | C2×#368073 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275269);