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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2220 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)(7,9)(8,10)(11,12)(13,14,19,23,20,24,18,32)(15,25,27,36,28,35,17,21)(16,22,30,38,31,39,33,37)(26,40,41,43,42,44,29,34);(5,6)(7,11)(8,10)(9,12)(13,15,20,28)(14,21,24,36)(16,26,31,42)(17,18,27,19)(22,34,39,43)(23,35,32,25)(29,33,41,30)(37,40,38,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,31)(14,22,24,39)(15,26,28,42)(17,29,27,41)(18,33,19,30)(21,34,36,43)(23,38,32,37)(25,40,35,44);(7,12)(9,11)(13,17,19,15,20,27,18,28)(14,21,23,25,24,36,32,35)(16,29,30,26,31,41,33,42)(22,34,38,40,39,43,37,44);(6,10)(9,11)(13,18,20,19)(14,23,24,32)(15,27,28,17)(16,30,31,33)(21,35,36,25)(22,37,39,38)(26,29,42,41)(34,40,43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6928824);