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×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(13,14,17,31,18,23,19,22)(15,24,32,36,27,20,26,35)(16,28,33,21,30,37,29,38)(25,40,41,39,42,44,43,34);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,32)(18,27)(19,26)(21,34)(22,36)(23,24)(28,44)(29,43)(30,42)(31,35)(33,41)(37,40)(38,39);(1,3,4,2)(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,34)(22,37)(23,38)(24,39)(26,41)(27,42)(28,31)(32,43)(35,44)(36,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,31)(15,26,27,32)(16,29,30,33)(20,35,24,36)(21,37,38,28)(25,41,42,43)(34,44,39,40);(7,11)(10,12)(15,27)(20,24)(25,42)(26,32)(34,39)(35,36)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,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, 6459851);