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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2153 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)(7,11)(8,12)(9,10)(13,14,18,19)(15,23,26,20)(16,27,30,21)(17,31,33,22)(24,34,28,36)(25,35,39,32)(29,37,43,40)(38,44,41,42);(2,3)(6,10)(11,12)(13,15,18,26)(14,20,19,23)(16,28,30,24)(17,32,33,35)(21,36,27,34)(22,25,31,39)(29,42,43,44)(37,41,40,38);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,30)(14,21,19,27)(15,24,26,28)(17,29,33,43)(20,34,23,36)(22,37,31,40)(25,38,39,41)(32,44,35,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,25)(16,29)(18,33)(19,31)(20,35)(21,37)(23,32)(24,38)(26,39)(27,40)(28,41)(30,43)(34,42)(36,44);(6,10)(11,12)(16,30)(17,33)(21,27)(22,31)(24,28)(25,39)(32,35)(34,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,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, 7468703);