Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,23,19,22,33,39)(15,24,27,35,26,40,17,20)(16,28,32,44,31,21,36,38)(25,42,43,34,29,41,30,37);(5,6)(7,10)(8,11)(9,12)(13,15,19,26)(14,20,22,35)(16,29,31,25)(17,33,27,18)(21,37,28,34)(23,40,39,24)(30,32,43,36)(38,41,44,42);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,31)(14,21,22,28)(15,25,26,29)(17,30,27,43)(18,32,33,36)(20,34,35,37)(23,38,39,44)(24,41,40,42);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,27,33,26)(14,20,23,24,22,35,39,40)(16,30,32,25,31,43,36,29)(21,34,38,41,28,37,44,42);(5,9,10,8)(6,11,7,12)(14,22)(15,26)(16,31)(17,27)(23,39)(32,36)(34,37)(41,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106373);