Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | |
2730 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)(15,20,18,31)(16,26,28,33)(17,22,30,38)(21,32,36,24)(25,34,29,37)(27,40,42,43)(35,41,44,39);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,29,30,25)(21,33,36,26)(22,37,38,34)(27,41,42,39)(35,40,44,43);(2,4)(13,16)(14,21)(15,24)(17,27)(18,32)(19,28)(20,33)(22,35)(23,36)(25,39)(26,31)(29,41)(30,42)(34,43)(37,40)(38,44);(6,10)(9,11)(13,17,19,30)(14,22,23,38)(15,25,18,29)(16,27,28,42)(20,34,31,37)(21,35,36,44)(24,39,32,41)(26,40,33,43);(2,4)(7,12)(9,11)(16,28)(17,30)(21,36)(22,38)(24,32)(25,29)(26,33)(34,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575298);