Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,22,16,20,14,21)(11,23,27,41,28,17,26,42)(12,29,33,45,34,18,32,46)(13,35,36,48,37,19,38,47)(24,39,52,57,53,49,30,43)(25,40,54,64,55,50,56,65)(31,44,60,67,61,58,62,68)(51,69,59,72,70,63,71,66);(2,5)(6,7)(9,11,16,28)(10,17,20,23)(12,30,34,52)(13,25,37,55)(14,27,15,26)(18,43,29,57)(19,40,35,50)(21,42,22,41)(24,33,53,32)(31,59,61,71)(36,56,38,54)(39,46,49,45)(44,66,58,72)(47,65,48,64)(51,62,70,60)(63,68,69,67);(9,12,16,34)(10,18,20,29)(11,24,28,53)(13,31,37,61)(14,33,15,32)(17,39,23,49)(19,44,35,58)(21,46,22,45)(25,51,55,70)(26,52,27,30)(36,62,38,60)(40,63,50,69)(41,57,42,43)(47,68,48,67)(54,71,56,59)(64,72,65,66);(1,3,4,8)(2,6,5,7)(9,13)(10,19)(11,25)(12,31)(14,36)(15,38)(16,37)(17,40)(18,44)(20,35)(21,47)(22,48)(23,50)(24,51)(26,54)(27,56)(28,55)(29,58)(30,59)(32,60)(33,62)(34,61)(39,63)(41,64)(42,65)(43,66)(45,67)(46,68)(49,69)(52,71)(53,70)(57,72)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268338);