Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3539 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,19,23)(15,24,27,33)(16,21,18,31)(17,22,29,37)(20,32,35,25)(26,38,40,43)(28,36,30,41)(34,42,44,39);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,39)(29,40)(30,42)(36,43)(37,44)(38,41);(2,4)(6,9)(7,11)(13,16,19,18)(14,21,23,31)(15,25,27,32)(17,28,29,30)(20,33,35,24)(22,36,37,41)(26,39,40,42)(34,43,44,38);(5,7,8,11)(6,10,9,12)(13,17,19,29)(14,22,23,37)(15,26,27,40)(16,28,18,30)(20,34,35,44)(21,36,31,41)(24,38,33,43)(25,39,32,42);(6,9)(10,12)(17,29)(22,37)(26,40)(28,30)(34,44)(36,41)(38,43)(39,42)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419882);