Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15312 |
| 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 | |
1271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,19,40)(4,10,23,36)(5,24,27,38)(6,14,8,12)(9,29,32,18)(11,26,39,28)(16,43,46,41)(17,31,47,55)(20,50,53,56)(21,35,22,34)(25,42,49,37)(30,48,59,45)(33,52,61,54)(44,58,63,64)(51,62,57,60);(1,3)(2,9)(4,16)(5,25)(6,29)(7,19)(8,18)(10,30)(11,37)(12,40)(13,32)(14,15)(17,26)(20,51)(21,48)(22,45)(23,46)(24,55)(27,49)(28,47)(31,38)(33,60)(34,41)(35,43)(36,59)(39,42)(44,52)(50,64)(53,57)(54,63)(56,58)(61,62);(1,4,8,21,7,23,6,22)(2,10,14,34,13,36,12,35)(3,16,18,48,19,46,29,45)(5,20,26,54,27,53,28,52)(9,30,15,41,32,59,40,43)(11,33,38,50,39,61,24,56)(17,44,49,51,47,63,25,57)(31,58,42,60,55,64,37,62);(1,5)(2,11)(3,17)(4,20)(6,28)(7,27)(8,26)(9,31)(10,33)(12,24)(13,39)(14,38)(15,42)(16,44)(18,49)(19,47)(21,54)(22,52)(23,53)(25,29)(30,58)(32,55)(34,50)(35,56)(36,61)(37,40)(41,60)(43,62)(45,57)(46,63)(48,51)(59,64)Polycyclic presentation:
2^9|a^2=i,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330377);