Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2394 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)(15,26)(16,32)(17,23)(18,24)(19,37)(20,25)(21,30)(22,33)(27,43)(28,40)(29,38)(31,44)(34,39)(35,41)(36,42);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,30,20,31,18,29)(14,21,24,26,25,38,37,44)(16,27,33,36,34,17,28,35)(22,23,32,42,40,43,39,41);(7,11)(9,12)(13,16,19,33,20,34,18,28)(14,22,24,32,25,40,37,39)(15,27,30,36,31,17,29,35)(21,23,26,42,38,43,44,41);(7,11)(9,12)(13,17,20,27)(14,23,25,43)(15,28,31,33)(16,29,34,30)(18,36,19,35)(21,32,38,39)(22,26,40,44)(24,42,37,41);(6,10)(9,12)(13,18,20,19)(14,24,25,37)(15,29,31,30)(16,33,34,28)(17,35,27,36)(21,26,38,44)(22,39,40,32)(23,41,43,42)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,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=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380527);