Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420074 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15274 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1316 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)(3,15,20,34)(4,21,25,42)(5,26,30,11)(6,12)(7,13)(8,14)(9,18,35,19)(10,24,38,31)(16,32)(17,46)(22,36)(23,49)(27,33)(28,41,29,40)(37,53)(39,48)(43,56,51,61)(44,57)(45,58)(47,59)(50,60,63,64)(52,55)(54,62);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,36)(11,39)(12,34)(13,15)(14,35)(16,24)(17,28)(21,32)(23,51)(25,44)(26,46)(29,48)(30,47)(31,45)(33,40)(37,61)(38,57)(41,59)(42,58)(43,54)(49,60)(50,52)(53,63)(55,56)(62,64);(1,4)(2,10)(3,16)(5,23)(6,24)(7,31)(8,25)(9,32)(11,37)(12,21)(13,42)(14,38)(15,36)(17,43)(18,22)(19,44)(20,45)(26,55)(27,50)(28,52)(29,53)(30,54)(33,56)(34,57)(35,58)(39,60)(40,49)(41,62)(46,64)(47,63)(48,51)(59,61);(1,5,6,28,8,30,7,29)(2,11,12,40,14,26,13,41)(3,17,18,27,20,48,19,47)(4,23,24,52,25,54,31,53)(9,33,15,39,35,59,34,46)(10,37,21,49,38,55,42,62)(16,43,22,50,45,51,44,63)(32,56,36,60,58,61,57,64)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327270);