Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 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)(16,29,31,36)(17,24,19,22)(21,30,37,32)(26,40,42,43)(27,35,33,38)(34,41,39,44);(2,3)(5,7,8,12)(6,9,10,11)(13,15,18,28)(14,20,23,25)(16,26,31,42)(17,33,19,27)(21,34,37,39)(22,38,24,35)(29,43,36,40)(30,44,32,41);(1,2,4,3)(6,10)(7,12)(13,16,17,30,18,31,19,32)(14,21,22,36,23,37,24,29)(15,26,33,44,28,42,27,41)(20,34,38,40,25,39,35,43);(6,10)(9,11)(13,17,18,19)(14,22,23,24)(15,27,28,33)(16,30,31,32)(20,35,25,38)(21,36,37,29)(26,41,42,44)(34,43,39,40);(1,3,4,2)(15,28)(20,25)(26,42)(27,33)(34,39)(35,38)(40,43)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,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=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723488);