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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14,19,24)(15,20,27,38)(16,22,30,40)(17,23,18,33)(21,32,41,29)(25,35,28,42)(26,36,31,37)(34,43,39,44);(2,3)(6,9)(7,11)(13,15,19,27)(14,20,24,38)(16,28,30,25)(17,31,18,26)(21,39,41,34)(22,42,40,35)(23,37,33,36)(29,43,32,44);(5,6)(7,10)(8,9)(11,12)(13,16,19,30)(14,21,24,41)(15,25,27,28)(17,29,18,32)(20,34,38,39)(22,23,40,33)(26,43,31,44)(35,37,42,36);(1,3,4,2)(6,9)(10,12)(13,16)(14,22)(15,25)(17,32)(18,29)(19,30)(20,35)(21,33)(23,41)(24,40)(26,44)(27,28)(31,43)(34,36)(37,39)(38,42);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,33)(15,26,27,31)(16,29,30,32)(20,36,38,37)(21,40,41,22)(25,43,28,44)(34,35,39,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,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, 7053223);