Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,9)(8,10)(11,12)(13,14,18,23)(15,19,25,34)(16,27,28,37)(17,21,31,22)(20,30,35,29)(24,38,39,43)(26,36,41,33)(32,42,44,40);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,24)(17,26)(18,25)(20,32)(21,36)(22,33)(23,34)(27,38)(28,39)(29,40)(30,42)(31,41)(35,44)(37,43);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,20)(15,24)(17,30)(18,28)(19,32)(21,27)(22,37)(23,35)(25,39)(26,42)(29,31)(33,43)(34,44)(36,38)(40,41);(6,7)(10,11)(14,21)(15,25)(17,31)(19,33)(20,27)(22,23)(24,39)(29,30)(32,43)(34,36)(35,37)(38,44);(16,28)(20,35)(24,39)(27,37)(29,30)(32,44)(38,43)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408962);