Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2209 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,20,18,24,17,23)(15,25,28,37,26,21,29,35)(16,30,34,40,33,22,31,36)(27,38,41,43,32,39,42,44);(5,6)(7,11)(8,10)(9,12)(14,20)(15,26)(16,31)(17,19)(21,37)(23,24)(25,35)(27,41)(32,42)(33,34)(36,40)(38,39);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,32)(17,29)(18,26)(19,28)(20,35)(22,39)(23,37)(24,25)(27,33)(30,38)(31,41)(34,42)(36,44)(40,43);(7,12)(9,11)(13,16,17,31,18,33,19,34)(14,22,23,40,24,30,20,36)(15,27,29,42,26,32,28,41)(21,38,37,44,25,39,35,43);(6,10)(9,11)(13,17,18,19)(14,23,24,20)(15,28,26,29)(16,31,33,34)(21,35,25,37)(22,40,30,36)(27,41,32,42)(38,43,39,44)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6929056);