Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23924 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14,19,23)(15,24,27,33)(16,21,18,31)(17,22,30,38)(20,32,35,25)(26,39,42,43)(28,36,29,37)(34,41,44,40);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,41)(30,42)(36,43)(37,39)(38,44);(2,4)(6,9)(10,12)(13,16,19,18)(14,21,23,31)(15,25,27,32)(17,29,30,28)(20,33,35,24)(22,37,38,36)(26,41,42,40)(34,39,44,43);(6,9)(10,12)(13,17)(14,22)(15,26)(16,28)(18,29)(19,30)(20,34)(21,36)(23,38)(24,39)(25,40)(27,42)(31,37)(32,41)(33,43)(35,44);(5,7,8,11)(6,10,9,12)(17,30)(22,38)(26,42)(28,29)(34,44)(36,37)(39,43)(40,41)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398106);