Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,25)(15,20,28,36)(16,29,32,38)(17,24,18,23)(21,30,39,31)(22,33,35,27)(26,37,41,44)(34,42,43,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,33)(18,27)(19,28)(21,34)(22,23)(24,35)(25,36)(29,37)(30,42)(31,40)(32,41)(38,44)(39,43);(2,4)(13,16,19,32)(14,21,25,39)(15,26,28,41)(17,31,18,30)(20,34,36,43)(22,37,35,44)(23,29,24,38)(27,42,33,40);(6,10)(7,12)(13,15)(14,22)(16,26)(17,27)(18,33)(19,28)(20,23)(21,37)(24,36)(25,35)(29,34)(30,40)(31,42)(32,41)(38,43)(39,44);(13,17,19,18)(14,23,25,24)(15,27,28,33)(16,30,32,31)(20,35,36,22)(21,38,39,29)(26,40,41,42)(34,37,43,44)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407924);