Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1312 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,25,29,17)(13,30,32,18)(14,20)(15,19)(21,47,49,36)(22,50,52,37)(23,39)(24,38)(26,54,53,40)(27,41)(28,46,34,42)(31,59,62,43)(33,45)(35,44)(48,63)(51,67,60,64)(55,58)(56,65)(57,66)(61,68)(69,70)(71,72);(1,3)(2,4)(5,8)(6,7)(9,11,15,24)(10,16,19,38)(12,26,29,53)(13,22,32,52)(14,33,35,23)(17,40,25,54)(18,37,30,50)(20,45,44,39)(21,34,49,28)(27,56,58,71)(31,60,62,51)(36,46,47,42)(41,65,55,72)(43,67,59,64)(48,61,70,57)(63,68,69,66);(1,4,6,5)(2,7,8,3)(9,12)(10,17)(11,21)(13,27)(14,34)(15,29)(16,36)(18,41)(19,25)(20,46)(22,48)(23,53)(24,49)(26,33)(28,35)(30,55)(31,61)(32,58)(37,63)(38,47)(39,54)(40,45)(42,44)(43,68)(50,69)(51,71)(52,70)(56,60)(57,62)(59,66)(64,72)(65,67);(1,5,6,4)(2,7,8,3)(9,13,15,32)(10,18,19,30)(11,22,24,52)(12,27,29,58)(14,31,35,62)(16,37,38,50)(17,41,25,55)(20,43,44,59)(21,48,49,70)(23,51,33,60)(26,56,53,71)(28,57,34,61)(36,63,47,69)(39,64,45,67)(40,65,54,72)(42,66,46,68)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290900);