Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,19,22,18,24,17,23)(15,20,26,35,28,36,27,34)(16,29,30,21,32,38,31,37)(25,39,41,33,42,43,40,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,24,36)(16,25,32,42)(17,27,19,26)(21,33,37,44)(22,35,23,34)(29,39,38,43)(30,41,31,40);(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,29)(23,38)(24,37)(26,40)(27,41)(28,42)(34,43)(35,39)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,24,23)(15,26,28,27)(16,30,32,31)(20,34,36,35)(21,29,37,38)(25,40,42,41)(33,43,44,39);(7,11)(10,12)(13,17,18,19)(14,22,24,23)(15,27,28,26)(16,30,32,31)(20,35,36,34)(21,29,37,38)(25,41,42,40)(33,39,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452292);