Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#466 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1072 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)(5,6,9,16,13,19,12,37)(7,20,25,41,26,48,22,51)(8,27,28,58,32,60,31,46)(10,18,11,36,33,49,35,17)(14,38,42,23,43,24,40,34)(15,44,45,62,47,29,59,30)(21,39,55,65,56,66,52,64)(50,61,57,63,67,53,68,54);(1,2,3,4)(5,7,10,34,12,25,35,38,13,26,33,23,9,22,11,24)(6,14,36,20,16,40,49,51,19,43,17,48,37,42,18,41)(8,21,29,61,31,55,62,63,32,56,44,53,28,52,30,54)(15,39,58,50,45,64,60,68,47,66,46,67,59,65,27,57);(5,8,10,29,12,31,35,62,13,32,33,44,9,28,11,30)(6,15,36,58,16,45,49,60,19,47,17,46,37,59,18,27)(7,21,34,61,25,55,38,63,26,56,23,53,22,52,24,54)(14,39,20,50,40,64,51,68,43,66,48,67,42,65,41,57);(5,9,13,12)(6,16,19,37)(7,22,26,25)(8,28,32,31)(10,11,33,35)(14,40,43,42)(15,45,47,59)(17,18,36,49)(20,51,48,41)(21,52,56,55)(23,38,34,24)(27,58,60,46)(29,30,44,62)(39,64,66,65)(50,68,67,57)(53,63,61,54)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=ei,g^b=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402428);