Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508310 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13313 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,33,46,65,71,13,18,32,45,15,20,35,48)(11,21,24,51,25,41,57,70,23,16,37,39,26,40,58,69)(12,27,30,49,36,66,64,43,29,60,63,44,31,47,34,17)(22,52,54,68,28,61,62,38,53,67,72,42,55,59,56,50);(2,5)(6,8)(9,11)(10,16)(12,28)(13,23)(14,24)(15,26)(17,42)(18,21)(19,39)(20,41)(22,31)(25,33)(27,59)(29,55)(30,62)(32,37)(34,54)(35,58)(36,53)(38,44)(40,46)(43,68)(45,51)(47,67)(48,70)(49,50)(52,66)(56,63)(57,65)(60,61)(64,72)(69,71);(3,7)(6,8)(9,12)(10,17)(11,22)(13,29)(14,34)(15,36)(16,38)(18,43)(19,47)(20,49)(21,50)(23,53)(24,56)(25,55)(26,28)(27,48)(30,35)(31,33)(32,64)(37,62)(39,61)(40,68)(41,42)(44,46)(45,66)(51,59)(52,69)(54,58)(57,72)(60,71)(63,65)(67,70);(1,3,4,7)(2,6,5,8)(12,29)(17,43)(22,53)(27,60)(28,55)(30,63)(31,36)(34,64)(38,50)(42,68)(44,49)(47,66)(52,67)(54,72)(56,62)(59,61)Polycyclic presentation:
2^9|a^2=f,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415595);