Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62825 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,10,6,9,20,24)(3,11,14,25,13,7,18,22)(4,15,16,26,17,8,19,23)(12,21,29,31,28,27,30,32)(33,34)(35,43)(36,47,49,40)(37,50)(38,42)(39,46)(41,53)(44,59,60,64)(45,57)(48,56,52,58)(51,55)(54,61,63,62);(1,3)(2,7)(4,12)(5,18)(6,13)(8,21)(9,11)(10,25)(14,20)(15,27)(16,30)(17,28)(19,29)(22,24)(23,32)(26,31)(33,35)(34,39)(36,44)(37,51)(38,45)(40,54)(41,57)(42,55)(43,50)(46,53)(47,63)(48,62)(49,60)(52,61)(56,59)(58,64);(1,4,6,17)(2,8,9,15)(3,12,13,28)(5,19,20,16)(7,21,11,27)(10,26,24,23)(14,30,18,29)(22,32,25,31)(33,36,38,49)(34,40,42,47)(35,44,45,60)(37,52,53,48)(39,54,55,63)(41,58,50,56)(43,59,57,64)(46,62,51,61);(2,9)(3,13)(8,15)(10,24)(12,28)(14,18)(23,26)(29,30)(35,45)(39,55)(43,57)(44,60)(46,51)(54,63)(59,64)(61,62)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94341);