Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1427 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,30,32,18)(14,19)(15,20)(17,28)(21,47)(22,50,52,37)(23,38)(24,39)(26,54)(27,55,58,41)(29,46)(31,59,62,43)(33,44)(34,45)(35,42)(36,49)(40,53)(48,69,70,63)(51,67,60,64)(56,72,71,65)(57,68,61,66);(2,5)(6,8)(9,11,14,23)(10,16,19,38)(12,26,28,53)(13,22,32,52)(15,34,33,24)(17,40,25,54)(18,37,30,50)(20,45,44,39)(21,35,49,29)(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);(3,7)(6,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,35)(16,36)(18,41)(19,25)(20,46)(22,48)(23,49)(24,53)(26,34)(29,33)(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);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,27)(14,32)(15,31)(16,37)(17,41)(19,30)(20,43)(21,48)(23,52)(24,51)(25,55)(26,56)(28,58)(29,57)(33,62)(34,60)(35,61)(36,63)(38,50)(39,64)(40,65)(42,66)(44,59)(45,67)(46,68)(47,69)(49,70)(53,71)(54,72)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362123);