Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1839 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)(7,10)(8,11)(9,12)(13,14,19,24,20,25,18,36)(15,26,28,23,29,42,17,21)(16,30,33,40,34,22,37,41)(27,44,32,38,31,43,35,39);(5,6)(7,10)(8,11)(9,12)(13,15,20,29)(14,21,25,23)(16,31,34,27)(17,18,28,19)(22,39,30,38)(24,42,36,26)(32,37,35,33)(40,44,41,43);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,22)(15,27)(17,35)(18,37)(19,33)(20,34)(21,38)(23,39)(24,41)(25,30)(26,43)(28,32)(29,31)(36,40)(42,44);(5,8,10,9)(6,11,7,12)(13,17,19,15,20,28,18,29)(14,23,24,42,25,21,36,26)(16,32,33,31,34,35,37,27)(22,38,41,43,30,39,40,44);(5,9,10,8)(6,11,7,12)(13,18,20,19)(14,24,25,36)(15,17,29,28)(16,33,34,37)(21,26,23,42)(22,40,30,41)(27,32,31,35)(38,44,39,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106168);