Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,18,19)(15,24,25,37)(16,28,31,21)(17,22,33,23)(20,35,34,27)(26,42,29,41)(30,40,32,39)(36,44,38,43);(2,3)(5,6)(7,10)(8,9)(11,12)(14,19)(15,25)(21,28)(22,23)(26,29)(27,35)(39,40)(43,44);(1,2,4,3)(6,9)(10,12)(13,15,17,27,18,25,33,35)(14,20,23,24,19,34,22,37)(16,29,32,43,31,26,30,44)(21,38,39,42,28,36,40,41);(5,7,8,11)(6,10,9,12)(13,16,18,31)(14,21,19,28)(15,26,25,29)(17,32,33,30)(20,36,34,38)(22,40,23,39)(24,41,37,42)(27,44,35,43);(1,3,4,2)(13,17,18,33)(14,22,19,23)(15,27,25,35)(16,30,31,32)(20,37,34,24)(21,39,28,40)(26,43,29,44)(36,41,38,42)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724496);