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 |
| prime radical | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,25)(16,31)(17,22)(18,23)(19,35)(20,24)(21,29)(26,43)(27,36)(28,38)(30,42)(32,37)(33,41)(34,40)(39,44);(6,9)(7,11)(13,15,20,30)(14,21,24,38)(16,32,17,26)(18,29,19,28)(22,39,31,36)(23,42,35,25)(27,33,44,34)(37,40,43,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,26)(17,20)(18,34)(19,33)(21,36)(23,41)(24,31)(25,37)(27,28)(29,44)(30,32)(35,40)(38,39)(42,43);(6,9)(10,12)(13,17,20,16)(14,22,24,31)(15,27,30,44)(18,34,19,33)(21,37,38,43)(23,40,35,41)(25,36,42,39)(26,29,32,28);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,24,35)(15,28,30,29)(16,33,17,34)(21,25,38,42)(22,40,31,41)(26,44,32,27)(36,43,39,37)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055661);