Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14,19,24)(15,20,28,36)(16,21,31,39)(17,22,18,23)(25,33,42,44)(26,34,27,35)(29,37,41,43)(30,38,32,40);(5,7,10,6)(8,12,9,11)(13,15)(14,20)(16,29)(17,26)(18,27)(19,28)(21,37)(22,34)(23,35)(24,36)(25,30)(31,41)(32,42)(33,38)(39,43)(40,44);(2,4)(13,16,19,31)(14,21,24,39)(15,25,28,42)(17,30,18,32)(20,33,36,44)(22,38,23,40)(26,41,27,29)(34,43,35,37);(5,8,10,9)(6,11,7,12)(13,17,19,18)(14,22,24,23)(15,26,28,27)(16,30,31,32)(20,34,36,35)(21,38,39,40)(25,41,42,29)(33,43,44,37);(5,9,10,8)(6,11,7,12)(15,27)(17,18)(20,35)(22,23)(25,29)(26,28)(30,32)(33,37)(34,36)(38,40)(41,42)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584062);