Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,20)(3,12,15,8,16,22,14,23)(4,9,17,25,19,26,18,24)(13,27,28,21,29,31,30,32)(33,34)(35,44)(36,41)(37,42)(38,51)(39,43)(40,46)(45,60)(47,54)(48,56)(49,58)(50,57)(52,55)(53,61)(59,62)(63,64);(1,3,6,15,7,16,5,14)(2,8,10,22,11,23,20,12)(4,13,17,28,19,29,18,30)(9,21,25,31,26,32,24,27)(33,35)(34,40)(36,45)(37,46)(38,52)(39,47)(41,53)(42,54)(43,55)(44,51)(48,61)(49,63)(50,62)(56,59)(57,64)(58,60);(1,4,7,19)(2,9,11,26)(3,13,16,29)(5,18,6,17)(8,21,23,32)(10,25,20,24)(12,27,22,31)(14,30,15,28)(33,36,39,50)(34,41,43,57)(35,45,47,62)(37,49,38,48)(40,53,55,64)(42,58,51,56)(44,59,54,60)(46,63,52,61);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,19,18)(8,22,23,12)(9,24,26,25)(13,28,29,30)(21,27,32,31)(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,48,50,49)(40,54,55,44)(41,56,57,58)(45,61,62,63)(53,59,64,60)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99489);