Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,19,25)(15,26,28,21)(16,22,31,40)(17,33,20,23)(18,24,34,36)(27,41,42,35)(29,39,43,37)(30,44,32,38);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,21,25,26)(16,29,31,43)(17,18,20,34)(22,37,40,39)(23,24,33,36)(27,30,42,32)(35,44,41,38);(1,2,3,4)(6,9)(10,12)(13,16,17,30,19,31,20,32)(14,22,33,44,25,40,23,38)(15,27,34,43,28,42,18,29)(21,35,24,39,26,41,36,37);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,33)(15,18,28,34)(16,30,31,32)(21,24,26,36)(22,38,40,44)(27,29,42,43)(35,39,41,37);(13,18,17,15,19,34,20,28)(14,24,33,26,25,36,23,21)(16,29,30,27,31,43,32,42)(22,39,44,41,40,37,38,35)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724454);