Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| 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 | #3497 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,25)(13,31)(14,35)(15,20)(17,29)(18,34)(19,27)(21,41)(22,51)(23,47)(24,40)(26,37)(28,45)(30,56)(32,42)(33,62)(36,39)(38,54)(43,48)(44,65)(46,59)(49,68)(50,70)(52,66)(53,72)(55,60)(57,69)(58,67)(61,64)(63,71);(2,6)(7,8)(9,11,15,24)(10,16,20,40)(12,26,30,58)(13,22,33,53)(14,36,27,23)(17,41,43,70)(18,44,46,38)(19,47,35,39)(21,48,50,29)(25,37,56,67)(28,57,32,63)(31,51,62,72)(34,65,59,54)(42,71,45,69)(49,64,52,60)(55,68,61,66);(1,3,5,4)(2,7,6,8)(9,12,27,48,15,30,14,29)(10,17,35,56,20,43,19,25)(11,21,36,58,24,50,23,26)(13,32,59,60,33,28,34,64)(16,37,47,70,40,67,39,41)(18,45,62,55,46,42,31,61)(22,52,65,57,53,49,54,63)(38,68,72,69,44,66,51,71);(9,13)(10,18,20,46)(11,22)(12,28)(14,34)(15,33)(16,38,40,44)(17,42,43,45)(19,31,35,62)(21,49)(23,54)(24,53)(25,55,56,61)(26,57)(27,59)(29,60)(30,32)(36,65)(37,66,67,68)(39,51,47,72)(41,69,70,71)(48,64)(50,52)(58,63);(1,4,5,3)(2,7,6,8)(13,33)(18,46)(22,53)(28,32)(31,62)(34,59)(38,44)(42,45)(49,52)(51,72)(54,65)(55,61)(57,63)(60,64)(66,68)(69,71)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=h,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,g^b=gi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521466);