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 | C2×2−1+4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,16)(12,25)(13,30)(14,33)(15,20)(17,28)(18,31)(19,37)(21,42)(22,52)(23,40)(24,47)(26,38)(27,43)(29,48)(32,49)(34,41)(35,44)(36,46)(39,54)(45,56)(50,63)(51,65)(53,66)(55,64)(57,60)(58,72)(59,67)(61,69)(62,68)(70,71);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,26)(13,22)(14,34)(15,23)(17,42)(18,45)(19,47)(20,40)(21,28)(24,37)(25,38)(27,59)(29,55)(30,52)(31,56)(32,53)(33,41)(35,51)(36,54)(39,46)(43,67)(44,65)(48,64)(49,66)(50,60)(57,63)(58,61)(62,71)(68,70)(69,72);(9,12)(10,17)(11,21)(13,27)(14,35)(15,29)(16,38)(18,43)(19,48)(20,44)(22,50)(23,51)(24,55)(25,33)(26,34)(28,37)(30,57)(31,62)(32,61)(36,60)(39,63)(40,64)(41,65)(42,47)(45,70)(46,69)(49,68)(52,67)(53,71)(54,58)(56,59)(66,72);(1,3,5,4)(2,7,8,6)(9,13,15,32)(10,18,20,46)(11,22,23,53)(12,27,29,61)(14,36,37,31)(16,39,40,45)(17,43,44,69)(19,49,33,30)(21,50,51,71)(24,56,34,54)(25,57,48,68)(26,58,55,59)(28,62,35,60)(38,63,64,70)(41,66,47,52)(42,67,65,72);(1,4,5,3)(2,7,8,6)(11,23)(16,40)(21,51)(22,53)(24,34)(26,55)(38,64)(39,45)(41,47)(42,65)(50,71)(52,66)(54,56)(58,59)(63,70)(67,72)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,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, 7430309);