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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 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)(6,10)(8,11)(13,14,19,23)(15,20,18,33)(16,26,29,42)(17,22,31,38)(21,24,36,28)(25,34,32,44)(27,41,30,43)(35,39,37,40);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15,19,18)(14,20,23,33)(16,24,29,28)(17,25,31,32)(21,26,36,42)(22,34,38,44)(27,39,30,40)(35,41,37,43);(5,7)(6,11)(8,10)(9,12)(13,16,15,24,19,29,18,28)(14,21,33,42,23,36,20,26)(17,30,25,40,31,27,32,39)(22,37,44,41,38,35,34,43);(2,4)(6,10)(7,12)(13,17)(14,22)(15,25)(16,27)(18,32)(19,31)(20,34)(21,35)(23,38)(24,39)(26,41)(28,40)(29,30)(33,44)(36,37)(42,43);(5,8,9,11)(6,7,10,12)(17,31)(22,38)(25,32)(27,30)(34,44)(35,37)(39,40)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092484);