Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6,10,30,11,16,23,15)(7,17,31,34,21,43,20,12)(8,13,35,37,26,38,25,56)(9,14,39,41,29,42,28,60)(18,45,47,62,22,44,51,32)(19,48,27,57,49,40,59,33)(24,52,54,65,55,36,63,67)(46,66,53,61,50,64,58,68);(5,7)(6,12)(8,22)(9,27)(10,31)(11,21)(13,32)(14,40)(15,43)(16,34)(17,30)(18,26)(19,28)(20,23)(24,53)(25,47)(29,59)(33,41)(35,51)(36,64)(37,45)(38,62)(39,49)(42,48)(44,56)(46,63)(50,54)(52,66)(55,58)(57,60)(61,65)(67,68);(2,4)(5,8,23,35,11,26,10,25)(6,13,30,56,16,38,15,37)(7,18,31,51,21,22,20,47)(9,24,39,63,29,55,28,54)(12,32,43,45,34,62,17,44)(14,36,60,67,42,52,41,65)(19,46,59,53,49,50,27,58)(33,61,48,66,57,68,40,64);(5,9,10,28,11,29,23,39)(6,14,15,41,16,42,30,60)(7,19,20,27,21,49,31,59)(8,24,25,54,26,55,35,63)(12,33,17,40,34,57,43,48)(13,36,37,65,38,52,56,67)(18,46,47,58,22,50,51,53)(32,61,44,64,62,68,45,66)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327691);