Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,22)(17,23)(18,20)(19,24)(25,36)(26,35)(27,34)(28,32)(29,38)(30,33)(31,37)(39,44)(40,43)(41,42);(2,4)(14,20)(15,25)(16,29)(17,19)(21,34)(22,37)(27,28)(30,31)(32,36)(33,38)(35,42)(39,40);(6,9)(10,12)(13,15,18,28)(14,21,20,32)(16,26,30,41)(17,25,19,27)(22,35,33,42)(23,36,24,34)(29,39,31,40)(37,43,38,44);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,29)(18,30)(19,31)(20,33)(21,35)(23,38)(24,37)(25,39)(27,40)(28,41)(32,42)(34,43)(36,44);(13,17,18,19)(14,23,20,24)(15,27,28,25)(16,29,30,31)(21,34,32,36)(22,38,33,37)(26,40,41,39)(35,43,42,44)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6356130);