Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45248 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,27,37)(16,22,29,38)(17,23,18,32)(21,31,39,28)(25,34,42,44)(26,35,30,36)(33,40,43,41);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,25)(17,30)(18,26)(19,27)(21,33)(22,34)(23,36)(24,37)(28,41)(29,42)(31,40)(32,35)(38,44)(39,43);(5,7,8,12)(6,9,10,11)(13,16,19,29)(14,21,24,39)(15,25,27,42)(17,28,18,31)(20,33,37,43)(22,23,38,32)(26,40,30,41)(34,36,44,35);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,25)(17,31)(18,28)(19,29)(20,34)(21,32)(23,39)(24,38)(26,41)(27,42)(30,40)(33,35)(36,43)(37,44);(6,10)(9,11)(13,17,19,18)(14,23,24,32)(15,26,27,30)(16,28,29,31)(20,35,37,36)(21,38,39,22)(25,40,42,41)(33,34,43,44)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6963312);