Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444406 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,22)(12,27)(13,19)(14,20)(15,37)(16,21)(17,25)(18,31)(23,43)(24,50)(26,41)(28,49)(29,54)(30,55)(32,45)(33,62)(34,47)(35,64)(36,48)(38,42)(39,57)(40,53)(44,60)(46,63)(51,67)(52,66)(56,68)(58,72)(59,70)(61,69)(65,71);(2,5)(6,8)(9,11)(10,17)(12,28)(13,33)(14,25)(15,26)(16,38)(18,43)(19,46)(20,22)(21,41)(23,32)(24,36)(27,39)(29,58)(30,57)(31,52)(34,63)(35,53)(37,42)(40,48)(44,68)(45,66)(47,62)(49,55)(50,64)(51,59)(54,67)(56,61)(60,71)(65,69)(70,72);(9,12,14,30)(10,18)(11,23)(13,29,34,59)(15,31,16,32)(17,39,22,49)(19,44)(20,45)(21,27)(24,51)(25,52)(26,28)(33,56)(35,60,36,61)(37,55)(38,57)(40,65,50,68)(41,66,42,43)(46,67,62,72)(47,69)(48,54)(53,58)(63,71)(64,70);(9,13,15,35,14,34,16,36)(10,19,37,64,20,47,21,48)(11,24,38,63,25,53,26,33)(12,29,31,60,30,59,32,61)(17,40,41,62,22,50,42,46)(18,44,55,70,45,69,27,54)(23,51,57,71,52,58,28,56)(39,65,66,72,49,68,43,67);(1,3,4,7)(2,6,5,8)(12,30)(18,45)(23,52)(27,55)(28,57)(29,59)(31,32)(39,49)(43,66)(44,69)(51,58)(54,70)(56,71)(60,61)(65,68)(67,72)Polycyclic presentation:
2^9|c^2=ghi,d^2=hi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373164);