Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2966 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)(7,11)(10,12)(13,14,17,22,18,21,31,37)(15,23,26,33,25,38,30,19)(16,20,29,36,28,35,43,44)(24,39,40,34,27,41,42,32);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,30)(18,25)(20,34)(21,33)(22,38)(23,37)(24,28)(26,31)(29,40)(32,35)(36,39)(41,44)(42,43);(2,4)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,35)(22,36)(23,39)(25,27)(26,40)(30,42)(31,43)(33,34)(37,44)(38,41);(6,9)(10,12)(15,25)(16,28)(19,33)(20,35)(23,38)(26,30)(29,43)(36,44);(5,7,8,11)(6,10,9,12)(14,21)(19,33)(20,35)(22,37)(23,38)(32,34)(36,44)(39,41)Polycyclic presentation:
2^9|a^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398380);