Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2935 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)(13,14,18,23,19,24,33,40)(15,25,28,36,29,39,32,20)(16,21,30,37,17,22,31,38)(26,41,43,35,27,42,44,34);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,24,36)(16,26,17,27)(18,32,33,28)(21,34,22,35)(23,39,40,25)(30,44,31,43)(37,42,38,41);(2,4)(6,9)(10,12)(13,16,19,17)(14,21,24,22)(15,26,29,27)(18,30,33,31)(20,34,36,35)(23,37,40,38)(25,41,39,42)(28,43,32,44);(6,9)(10,12)(13,17,19,16)(14,22,24,21)(15,27,29,26)(18,31,33,30)(20,35,36,34)(23,38,40,37)(25,42,39,41)(28,44,32,43);(5,7,8,11)(6,10,9,12)(16,17)(21,22)(26,27)(30,31)(34,35)(37,38)(41,42)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398446);