Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,25)(15,26,30,21)(16,31,33,22)(17,34,20,23)(18,37,32,24)(27,39,36,44)(28,40,38,42)(29,41,35,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,30)(14,21,25,26)(16,32,33,18)(17,28,20,38)(22,24,31,37)(23,40,34,42)(27,35,36,29)(39,41,44,43);(2,4)(13,16,17,35,19,33,20,29)(14,22,34,41,25,31,23,43)(15,27,38,18,30,36,28,32)(21,39,40,37,26,44,42,24);(6,10)(9,11)(13,17,19,20)(14,23,25,34)(15,28,30,38)(16,29,33,35)(18,36,32,27)(21,40,26,42)(22,41,31,43)(24,44,37,39);(7,12)(9,11)(13,18,20,27,19,32,17,36)(14,24,23,44,25,37,34,39)(15,29,28,33,30,35,38,16)(21,41,42,22,26,43,40,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6713913);