Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,18,32)(16,26,28,39)(17,22,30,36)(21,24,35,29)(25,33,31,42)(27,38,41,44)(34,37,43,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,32)(16,24,28,29)(17,25,30,31)(21,26,35,39)(22,33,36,42)(27,37,41,40)(34,38,43,44);(2,4)(13,16,15,24,19,28,18,29)(14,21,32,39,23,35,20,26)(17,27,25,37,30,41,31,40)(22,34,42,44,36,43,33,38);(6,10)(9,11)(13,17)(14,22)(15,25)(16,27)(18,31)(19,30)(20,33)(21,34)(23,36)(24,37)(26,38)(28,41)(29,40)(32,42)(35,43)(39,44);(2,4)(7,12)(9,11)(16,28)(17,30)(21,35)(22,36)(24,29)(25,31)(26,39)(33,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589719);