Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,22)(17,23)(18,21)(19,24)(25,35)(26,34)(27,36)(28,33)(29,38)(30,39)(31,37)(32,40)(41,43)(42,44);(6,9)(10,12)(13,15,18,27)(14,20,21,36)(16,29,31,41)(17,28,19,25)(22,38,37,43)(23,33,24,35)(26,30,42,32)(34,39,44,40);(1,2,3,4)(14,21)(15,25)(17,19)(20,33)(22,37)(26,41)(27,28)(29,42)(30,32)(34,38)(35,36)(43,44);(13,16,18,31)(14,22,21,37)(15,26,27,42)(17,30,19,32)(20,34,36,44)(23,39,24,40)(25,41,28,29)(33,38,35,43);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,21,24)(15,25,27,28)(16,30,31,32)(20,35,36,33)(22,39,37,40)(26,41,42,29)(34,43,44,38)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352530);