Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,25)(24,27)(26,30)(28,31)(29,32)(33,35)(34,36);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,32,28,34,27,33)(22,29,24,25,31,35,30,36);(7,13)(10,17)(14,19)(18,20)(21,24,26,31,28,30,27,22)(23,25,32,35,34,36,33,29);(7,13)(10,17)(14,19)(18,20)(21,25)(22,23)(24,32)(26,35)(27,29)(28,36)(30,33)(31,34);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,30,31,24)(23,32,34,33)(25,29,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=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, 6404940);