Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | #3487 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1852 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)(11,16)(12,25)(13,30)(14,33)(15,20)(17,28)(18,31)(19,37)(21,50)(22,54)(23,40)(24,47)(26,58)(27,43)(29,48)(32,49)(34,41)(35,44)(36,46)(38,53)(39,56)(42,52)(45,57)(51,65)(55,66)(59,62)(60,71)(61,67)(63,69)(64,68)(70,72);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,34)(15,23)(17,42)(18,45)(19,47)(20,40)(21,35)(24,37)(25,58)(27,61)(28,52)(29,53)(30,54)(31,57)(32,55)(33,41)(36,56)(38,48)(39,46)(43,67)(44,50)(49,66)(51,64)(59,70)(60,63)(62,72)(65,68)(69,71);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,27)(14,35)(15,29)(16,38)(18,43)(19,48)(20,44)(22,51)(23,52)(24,26)(25,33)(28,37)(30,59)(31,64)(32,63)(34,53)(36,62)(39,65)(40,58)(41,42)(45,70)(46,69)(47,50)(49,68)(54,71)(55,72)(56,61)(57,60)(66,67);(9,13,15,32)(10,18,20,46)(11,22,23,55)(12,27,29,63)(14,36,37,31)(16,39,40,45)(17,43,44,69)(19,49,33,30)(21,51,52,72)(24,57,34,56)(25,59,48,68)(26,60,53,61)(28,64,35,62)(38,65,58,70)(41,66,47,54)(42,67,50,71);(2,5)(6,8)(11,23)(16,40)(21,52)(22,55)(24,34)(26,53)(38,58)(39,45)(41,47)(42,50)(51,72)(54,66)(56,57)(60,61)(65,70)(67,71)Polycyclic presentation:
2^9|c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422111);