Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6,10,16,12,17,11,32)(7,18,22,45,21,13,23,36)(8,24,26,39,27,38,31,14)(9,15,29,41,30,43,33,42)(19,46,49,34,48,53,51,44)(20,40,50,61,28,35,52,60)(25,54,56,63,58,64,57,37)(47,65,66,59,55,68,67,62);(5,7,11,23,12,21,10,22)(6,13,16,36,17,18,32,45)(8,19,26,49,27,48,31,51)(9,28,29,52,30,20,33,50)(14,34,38,46,39,44,24,53)(15,40,42,60,43,35,41,61)(25,55,57,66,58,47,56,67)(37,62,54,65,63,59,64,68);(1,2,4,3)(5,8,12,27)(6,14,17,39)(7,19,21,48)(9,25,30,58)(10,31,11,26)(13,34,18,44)(15,37,43,63)(16,38,32,24)(20,47,28,55)(22,51,23,49)(29,57,33,56)(35,59,40,62)(36,46,45,53)(41,64,42,54)(50,67,52,66)(60,65,61,68);(1,3,4,2)(5,9)(6,15)(7,20)(8,25)(10,29)(11,33)(12,30)(13,35)(14,37)(16,41)(17,43)(18,40)(19,47)(21,28)(22,50)(23,52)(24,54)(26,56)(27,58)(31,57)(32,42)(34,59)(36,60)(38,64)(39,63)(44,62)(45,61)(46,65)(48,55)(49,66)(51,67)(53,68)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,f^c=fi,g^c=gi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264162);