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 | #24244 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2065 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)(13,14,19,24)(15,20,28,36)(16,21,31,39)(17,22,18,23)(25,33,27,35)(26,34,41,43)(29,37,42,44)(30,38,32,40);(6,9)(10,12)(13,15)(14,20)(16,29)(17,27)(18,25)(19,28)(21,37)(22,35)(23,33)(24,36)(26,30)(31,42)(32,41)(34,38)(39,44)(40,43);(1,2,3,4)(15,25)(17,18)(20,33)(22,23)(26,29)(27,28)(30,32)(34,37)(35,36)(38,40)(41,42)(43,44);(13,16,19,31)(14,21,24,39)(15,26,28,41)(17,30,18,32)(20,34,36,43)(22,38,23,40)(25,29,27,42)(33,37,35,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,25,28,27)(16,30,31,32)(20,33,36,35)(21,38,39,40)(26,29,41,42)(34,37,43,44)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,e^2=fi,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, 6352470);