Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,28,20)(16,29,31,38)(17,24,19,22)(21,33,36,30)(26,40,42,44)(27,35,32,37)(34,43,39,41);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,32)(18,28)(19,27)(21,34)(22,37)(23,25)(24,35)(29,44)(30,43)(31,42)(33,41)(36,39)(38,40);(1,2,4,3)(7,12)(9,11)(13,16,18,31)(14,21,23,36)(15,26,28,42)(17,30,19,33)(20,34,25,39)(22,29,24,38)(27,41,32,43)(35,40,37,44);(6,10)(9,11)(13,17,18,19)(14,22,23,24)(15,27,28,32)(16,30,31,33)(20,35,25,37)(21,29,36,38)(26,41,42,43)(34,40,39,44);(1,3,4,2)(15,28)(20,25)(26,42)(27,32)(34,39)(35,37)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717187);