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 | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2175 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)(17,31,34,22)(24,35,28,37)(25,36,42,32)(29,38,33,39)(40,44,41,43);(2,3)(6,10)(11,12)(13,15,18,26)(14,20,19,23)(16,28,30,24)(17,32,34,36)(21,37,27,35)(22,25,31,42)(29,43,33,44)(38,41,39,40);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,18,30)(14,21,19,27)(15,24,26,28)(17,33,34,29)(20,35,23,37)(22,39,31,38)(25,41,42,40)(32,43,36,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,18,34)(14,22,19,31)(15,25,26,42)(16,29,30,33)(20,36,23,32)(21,38,27,39)(24,40,28,41)(35,43,37,44);(6,10)(11,12)(16,30)(17,34)(21,27)(22,31)(24,28)(25,42)(32,36)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7468961);