Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186976 |
| 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 | #62831 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,5,10)(3,13,16,23,17,8,15,24)(4,18,19,26,20,9,21,25)(14,22,28,31,29,27,30,32)(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);(1,3,7,17)(2,8,11,13)(4,14,20,29)(5,16,6,15)(9,22,18,27)(10,24,12,23)(19,30,21,28)(25,32,26,31)(33,35,38,45)(34,39,42,55)(36,44,49,60)(37,51,53,46)(40,54,47,63)(41,57,50,43)(48,62,52,61)(56,59,58,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,28)(16,30)(17,29)(23,31)(24,32)(33,36)(34,40)(35,44)(37,52)(38,49)(39,54)(41,58)(42,47)(43,59)(45,60)(46,62)(48,53)(50,56)(51,61)(55,63)(57,64);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,20,21)(8,23,13,24)(9,25,18,26)(14,28,29,30)(22,31,27,32)(35,45)(39,55)(43,57)(44,60)(46,51)(54,63)(59,64)(61,62)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94324);