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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 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)(7,11)(10,12)(13,14,18,24)(15,25,29,20)(16,22,31,40)(17,23,19,34)(21,33,41,30)(26,36,27,42)(28,38,32,37)(35,44,39,43);(1,2)(3,4)(6,9)(7,11)(13,15,18,29)(14,20,24,25)(16,27,31,26)(17,32,19,28)(21,39,41,35)(22,36,40,42)(23,38,34,37)(30,43,33,44);(5,6)(7,10)(8,9)(11,12)(13,16,18,31)(14,21,24,41)(15,26,29,27)(17,30,19,33)(20,35,25,39)(22,23,40,34)(28,43,32,44)(36,37,42,38);(2,4)(6,9)(10,12)(13,16)(14,22)(15,27)(17,33)(18,31)(19,30)(20,36)(21,34)(23,41)(24,40)(25,42)(26,29)(28,43)(32,44)(35,38)(37,39);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,24,34)(15,28,29,32)(16,30,31,33)(20,37,25,38)(21,40,41,22)(26,43,27,44)(35,42,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7050211);