Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,27,35)(16,21,29,38)(17,30,18,22)(24,32,41,44)(25,33,26,34)(28,37,31,36)(39,42,40,43);(2,3)(6,9)(10,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,27)(21,32)(22,33)(23,35)(28,39)(29,41)(30,34)(31,40)(36,43)(37,42)(38,44);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,31)(18,28)(19,29)(20,32)(22,37)(23,38)(25,39)(26,40)(27,41)(30,36)(33,42)(34,43)(35,44);(1,3,4,2)(15,25)(17,18)(20,33)(22,30)(24,39)(26,27)(28,31)(32,42)(34,35)(36,37)(40,41)(43,44);(6,9)(10,12)(13,17,19,18)(14,22,23,30)(15,26,27,25)(16,28,29,31)(20,33,35,34)(21,36,38,37)(24,39,41,40)(32,43,44,42)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467604);