Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6)(7,17,20,12)(8,13)(9,25)(10,15)(11,29)(14,28)(16,26)(18,30,42,37)(19,44,46,52)(21,32,45,39)(22,48)(23,34)(24,51)(27,53)(31,47,38,60)(33,50)(35,40)(36,54)(41,63,64,66)(43,59,61,56)(49,67)(55,68,58,65)(57,62);(5,7,11,21,10,20,26,45)(6,12,29,39,15,17,16,32)(8,18,40,61,23,42,24,43)(9,19,28,47,27,46,54,60)(13,30,35,56,34,37,51,59)(14,31,53,44,36,38,25,52)(22,41,62,68,49,64,50,65)(33,55,48,63,57,58,67,66);(1,2,3,4)(5,8,10,23)(6,13,15,34)(7,18,20,42)(9,22,27,49)(11,24,26,40)(12,30,17,37)(14,33,36,57)(16,35,29,51)(19,41,46,64)(21,43,45,61)(25,48,53,67)(28,50,54,62)(31,55,38,58)(32,56,39,59)(44,66,52,63)(47,65,60,68);(5,9,26,54,10,27,11,28)(6,14,29,53,15,36,16,25)(7,19,45,60,20,46,21,47)(8,22,40,62,23,49,24,50)(12,31,39,44,17,38,32,52)(13,33,51,67,34,57,35,48)(18,41,61,68,42,64,43,65)(30,55,59,66,37,58,56,63)Polycyclic presentation:
2^9|a^2=gi,b^2=eh,c^2=fg,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324412);