Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16252 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1826 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)(3,14)(4,10)(5,25)(6,30)(7,12)(8,13)(9,17)(11,29)(15,34)(16,47)(18,43)(19,33)(20,52)(21,35)(22,40)(23,36)(24,37)(26,57)(27,39)(28,58)(31,45)(32,51)(38,54)(41,46)(42,50)(44,60)(48,59)(49,56)(53,62)(55,61)(63,64);(1,3,6,17,8,18,7,19)(2,9,12,14,13,33,30,43)(4,20,22,15,24,45,23,46)(5,16,27,49,29,50,28,51)(10,34,36,52,37,41,40,31)(11,32,39,42,25,56,58,47)(21,53,26,44,55,48,54,64)(35,60,38,62,61,63,57,59);(1,4)(2,10)(3,15)(5,26)(6,23)(7,22)(8,24)(9,31)(11,38)(12,40)(13,37)(14,41)(16,48)(17,20)(18,46)(19,45)(21,27)(25,57)(28,55)(29,54)(30,36)(32,60)(33,52)(34,43)(35,39)(42,59)(44,49)(47,62)(50,53)(51,64)(56,63)(58,61);(1,5,6,27,8,29,7,28)(2,11,12,39,13,25,30,58)(3,16,17,49,18,50,19,51)(4,21,22,26,24,55,23,54)(9,32,14,42,33,56,43,47)(10,35,36,38,37,61,40,57)(15,44,45,48,46,64,20,53)(31,59,34,60,52,62,41,63)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72264906, #72264914, #72265017, … |
| maximal normal subgroup of | #72264906, #72264914, #72265017, … |
| maximal quotient of | #72265248 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361630);