Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#3547 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2982 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)(13,14,19,22,18,24,17,23)(15,25,27,20,29,34,28,35)(16,21,30,37,32,38,31,36)(26,39,42,33,41,43,40,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,27)(18,29)(19,28)(21,33)(22,25)(23,34)(24,35)(30,40)(31,42)(32,41)(36,43)(37,39)(38,44);(2,4)(6,9)(10,12)(13,16,18,32)(14,21,24,38)(15,26,29,41)(17,31,19,30)(20,33,35,44)(22,37,23,36)(25,39,34,43)(27,42,28,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,24,23)(15,27,29,28)(16,30,32,31)(20,25,35,34)(21,36,38,37)(26,40,41,42)(33,43,44,39);(7,11)(10,12)(13,17,18,19)(14,23,24,22)(15,27,29,28)(16,30,32,31)(20,34,35,25)(21,37,38,36)(26,40,41,42)(33,39,44,43)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449876);