Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2332 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,31,38)(16,32,34,40)(17,23,37,44)(18,24,19,25)(21,30,39,29)(22,33,41,28)(35,42,36,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,29,20,31,19,30)(14,21,25,27,26,39,24,38)(16,17,33,36,34,37,28,35)(22,23,40,42,41,44,32,43);(7,11)(9,12)(13,16,18,33,20,34,19,28)(14,22,25,40,26,41,24,32)(15,17,29,36,31,37,30,35)(21,23,27,42,39,44,38,43);(7,11)(9,12)(13,17)(14,23)(15,28)(16,29)(18,36)(19,35)(20,37)(21,32)(22,27)(24,43)(25,42)(26,44)(30,34)(31,33)(38,41)(39,40);(6,10)(9,12)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,28,34,33)(17,35,37,36)(21,38,39,27)(22,40,41,32)(23,42,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380536);