Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1846 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,25,31)(23,30,26,27)(24,29,34,35)(28,33,36,32);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,25,26)(22,27,31,30)(24,32,34,33)(28,29,36,35);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,28)(23,32)(25,34)(26,33)(27,29)(30,35)(31,36);(7,14)(10,18)(13,15)(17,19)(21,24)(22,29)(23,33)(25,34)(26,32)(27,28)(30,36)(31,35);(6,12)(10,18)(11,20)(17,19)(21,23,25,26)(22,30,31,27)(24,32,34,33)(28,35,36,29)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384877);