Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1996 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)(13,14,18,22)(15,23,26,39)(16,20,29,36)(17,31,24,21)(19,32,34,27)(25,38,28,42)(30,44,40,37)(33,43,35,41);(5,6)(7,10)(8,9)(11,12)(13,15,24,32,18,26,17,27)(14,19,31,39,22,34,21,23)(16,28,40,41,29,25,30,43)(20,35,44,38,36,33,37,42);(2,4)(6,9)(7,11)(13,16)(14,20)(15,25)(17,30)(18,29)(19,33)(21,37)(22,36)(23,38)(24,40)(26,28)(27,41)(31,44)(32,43)(34,35)(39,42);(5,7,8,11)(6,10,9,12)(15,26)(19,34)(23,39)(25,28)(27,32)(33,35)(38,42)(41,43);(6,9)(10,12)(15,26)(16,29)(19,34)(20,36)(23,39)(27,32)(30,40)(37,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420752);