Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,27,34)(16,21,18,23)(17,22,30,37)(25,32,31,38)(26,33,41,44)(28,35,29,36)(39,42,40,43);(5,7)(6,9)(8,12)(10,11)(15,25)(16,18)(17,28)(20,32)(21,23)(22,35)(27,31)(29,30)(34,38)(36,37)(39,40)(42,43);(1,2,3,4)(13,15,19,27)(14,20,24,34)(16,25,18,31)(17,26,30,41)(21,32,23,38)(22,33,37,44)(28,39,29,40)(35,42,36,43);(6,10)(9,11)(13,16,19,18)(14,21,24,23)(15,25,27,31)(17,29,30,28)(20,32,34,38)(22,36,37,35)(26,40,41,39)(33,43,44,42);(7,12)(9,11)(13,17,19,30)(14,22,24,37)(15,26,27,41)(16,28,18,29)(20,33,34,44)(21,35,23,36)(25,39,31,40)(32,42,38,43)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589453);