Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2108 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)(13,14,18,22)(15,19,24,32)(16,21,27,36)(17,20,29,23)(25,31,37,43)(26,33,38,30)(28,34,40,35)(39,42,44,41);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,33)(21,31)(22,32)(23,30)(27,37)(28,39)(29,38)(34,42)(35,41)(36,43)(40,44);(5,7,8,12)(6,9,10,11)(14,20)(15,24)(16,27)(17,29)(19,30)(21,35)(22,23)(31,42)(32,33)(34,36)(39,44)(41,43);(2,4)(7,12)(9,11)(13,16)(14,21)(15,25)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(24,37)(26,39)(29,40)(30,41)(32,43)(33,42)(38,44);(6,10)(9,11)(14,22)(16,27)(19,32)(20,23)(25,37)(28,40)(30,33)(39,44)Polycyclic presentation:
2^9|a^2=i,c^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961259);