Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2673 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,21)(15,24,28,38)(16,22,30,35)(17,23,18,32)(20,26,33,27)(25,39,41,44)(29,36,31,37)(34,42,43,40);(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,28,17,26)(14,20,23,24,21,33,32,38)(16,25,29,40,30,41,31,42)(22,34,37,39,35,43,36,44);(1,2,3,4)(7,11)(9,12)(14,21)(20,33)(22,35)(23,32)(24,38)(34,43)(36,37)(39,44);(6,10)(9,12)(13,16)(14,22)(15,25)(17,31)(18,29)(19,30)(20,34)(21,35)(23,37)(24,39)(26,42)(27,40)(28,41)(32,36)(33,43)(38,44);(13,17,19,18)(14,23,21,32)(15,26,28,27)(16,29,30,31)(20,24,33,38)(22,36,35,37)(25,40,41,42)(34,44,43,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341005);