Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2965 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)(13,14,19,32,18,23,16,21)(15,24,25,20,28,33,27,35)(17,22,30,37,31,38,29,36)(26,39,40,34,42,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,35)(24,32)(29,40)(30,41)(31,42)(36,43)(37,39)(38,44);(2,4)(6,9)(10,12)(13,16,18,19)(14,21,23,32)(15,25,28,27)(17,30,31,29)(20,33,35,24)(22,37,38,36)(26,41,42,40)(34,39,44,43);(6,9)(10,12)(13,17,18,31)(14,22,23,38)(15,26,28,42)(16,29,19,30)(20,34,35,44)(21,36,32,37)(24,39,33,43)(25,40,27,41);(5,7,8,11)(6,10,9,12)(17,31)(22,38)(26,42)(29,30)(34,44)(36,37)(39,43)(40,41)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398431);