Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,23)(12,18)(13,33)(14,22)(15,20)(16,21)(17,26)(19,37)(24,50)(25,54)(27,41)(28,42)(29,58)(30,60)(31,49)(32,45)(34,46)(35,48)(36,47)(38,44)(39,57)(40,56)(43,62)(51,55)(52,71)(53,66)(59,65)(61,70)(63,69)(64,68)(67,72);(1,3,5,4)(2,6,8,7)(9,11,14,26,16,27,15,28)(10,17,20,23,21,42,22,41)(12,29,31,57,32,24,38,53)(13,34,35,56,36,55,37,25)(18,39,44,58,45,66,49,50)(19,46,47,54,48,51,33,40)(30,61,62,52,63,72,64,59)(43,67,60,71,68,70,69,65);(1,3,5,4)(2,7,8,6)(9,12)(10,18)(11,24)(13,30)(14,38)(15,31)(16,32)(17,39)(19,43)(20,49)(21,45)(22,44)(23,50)(25,52)(26,57)(27,29)(28,53)(33,60)(34,61)(35,64)(36,63)(37,62)(40,65)(41,58)(42,66)(46,70)(47,69)(48,68)(51,67)(54,71)(55,72)(56,59);(1,4,5,3)(2,7,8,6)(9,13,16,36)(10,19,21,48)(11,25,27,56)(12,30,32,63)(14,37,15,35)(17,40,42,54)(18,43,45,68)(20,33,22,47)(23,51,41,46)(24,52,29,59)(26,55,28,34)(31,64,38,62)(39,65,66,71)(44,69,49,60)(50,67,58,70)(53,61,57,72)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339449);