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 | #61076 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 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)(17,29)(18,32)(19,37)(21,42)(22,52)(23,47)(24,41)(26,38)(27,43)(28,48)(31,49)(34,40)(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);(2,5)(6,8)(9,11,15,24)(10,16,20,41)(12,26,28,55)(13,22,31,53)(14,34,37,23)(17,42,44,65)(18,45,46,39)(19,47,33,40)(21,35,51,29)(25,38,48,64)(27,59,61,58)(30,52,49,66)(32,56,36,54)(43,67,69,72)(50,62,71,60)(57,63,68,70);(9,12,15,28)(10,17,20,44)(11,21,24,51)(13,27,31,61)(14,35,37,29)(16,38,41,64)(18,43,46,69)(19,48,33,25)(22,50,53,71)(23,55,34,26)(30,57,49,68)(32,62,36,60)(39,63,45,70)(40,65,47,42)(52,67,66,72)(54,58,56,59);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,27)(14,36)(15,31)(16,39)(17,43)(19,49)(20,46)(21,50)(23,56)(24,53)(25,57)(26,58)(28,61)(29,62)(30,33)(32,37)(34,54)(35,60)(38,63)(40,66)(41,45)(42,67)(44,69)(47,52)(48,68)(51,71)(55,59)(64,70)(65,72);(2,5)(6,8)(12,28)(13,31)(17,44)(18,46)(21,51)(22,53)(25,48)(26,55)(29,35)(30,49)(32,36)(38,64)(39,45)(42,65)(52,66)(54,56)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^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,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423756);