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 | |
1817 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)(13,14,20,35,19,25,18,24)(15,26,28,21,29,37,17,23)(16,22,31,38,33,40,32,39)(27,42,30,36,43,44,34,41);(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,27)(17,20)(18,28)(19,29)(22,36)(23,25)(24,37)(26,35)(30,32)(31,34)(33,43)(38,42)(39,44)(40,41);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,22)(15,27)(17,34)(18,32)(19,33)(20,31)(21,36)(23,41)(24,39)(25,40)(26,42)(28,30)(29,43)(35,38)(37,44);(5,8,10,9)(6,11,7,12)(13,17,20,15,19,28,18,29)(14,23,35,26,25,21,24,37)(16,30,31,43,33,34,32,27)(22,36,38,44,40,41,39,42);(5,9,10,8)(6,11,7,12)(13,18,19,20)(14,24,25,35)(15,28,29,17)(16,31,33,32)(21,37,23,26)(22,38,40,39)(27,34,43,30)(36,42,41,44)Polycyclic presentation:
2^9|a^2=fi,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, 7106381);