Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(13,14)(15,20)(16,28)(17,23)(18,31)(19,22)(21,29)(24,38)(25,35)(26,34)(27,41)(30,37)(32,36)(33,40)(39,43)(42,44);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,33)(22,35)(23,34)(28,38)(29,40)(30,39)(31,41)(32,42)(36,44)(37,43);(1,3,4,2)(6,9)(7,11)(13,16,19,30)(14,21,22,36)(15,24,25,39)(17,29,18,32)(20,33,35,44)(23,37,31,28)(26,40,27,42)(34,43,41,38);(5,7)(6,10)(8,11)(9,12)(14,22)(15,25)(21,36)(23,31)(24,39)(26,27)(28,37)(40,42);(6,9)(7,11)(13,17,19,18)(14,23,22,31)(15,26,25,27)(16,29,30,32)(20,34,35,41)(21,37,36,28)(24,40,39,42)(33,43,44,38)Polycyclic presentation:
2^9|b^2=fh,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505581);