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)(13,14,19,23)(15,20,26,36)(16,27,30,39)(17,22,18,32)(21,33,40,29)(24,41,28,44)(25,35,31,37)(34,42,38,43);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,28)(17,31)(18,25)(19,26)(21,38)(22,37)(23,36)(24,30)(27,44)(29,42)(32,35)(33,43)(34,40)(39,41);(1,2,4,3)(7,12)(9,11)(13,16)(14,21)(15,24)(17,29)(18,33)(19,30)(20,34)(22,39)(23,40)(25,42)(26,28)(27,32)(31,43)(35,41)(36,38)(37,44);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,29,30,33)(20,35,36,37)(21,39,40,27)(24,42,28,43)(34,41,38,44);(1,3,4,2)(15,26)(20,36)(24,28)(25,31)(34,38)(35,37)(41,44)(42,43)Polycyclic presentation:
2^9|a^2=i,c^2=fhi,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, 6717188);