Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,23,26,20)(16,21,29,32)(17,30,31,22)(24,27,38,33)(25,34,39,36)(28,41,42,35)(37,43,44,40);(6,9)(7,11)(13,15,18,26)(14,20,19,23)(16,27,29,33)(17,25,31,39)(21,24,32,38)(22,34,30,36)(28,40,42,43)(35,37,41,44);(2,4)(13,16,18,29)(14,21,19,32)(15,24,26,38)(17,28,31,42)(20,33,23,27)(22,35,30,41)(25,37,39,44)(34,43,36,40);(6,9)(10,12)(13,17)(14,22)(15,25)(16,28)(18,31)(19,30)(20,34)(21,35)(23,36)(24,37)(26,39)(27,40)(29,42)(32,41)(33,43)(38,44);(5,7,8,11)(6,10,9,12)(14,19)(15,20)(17,31)(21,32)(23,26)(24,33)(25,36)(27,38)(28,42)(34,39)(37,40)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580213);