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 | #45093 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2151 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,18,32)(16,26,29,33)(17,22,30,37)(21,27,36,24)(25,34,31,42)(28,39,41,43)(35,40,44,38);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,23,32)(16,27,29,24)(17,25,30,31)(21,26,36,33)(22,34,37,42)(28,40,41,38)(35,39,44,43);(5,7,8,12)(6,9,10,11)(13,16,19,29)(14,21,23,36)(15,24,18,27)(17,28,30,41)(20,33,32,26)(22,35,37,44)(25,38,31,40)(34,43,42,39);(2,4)(7,12)(9,11)(13,17)(14,22)(15,25)(16,28)(18,31)(19,30)(20,34)(21,35)(23,37)(24,38)(26,39)(27,40)(29,41)(32,42)(33,43)(36,44);(6,10)(9,11)(14,23)(16,29)(17,30)(20,32)(24,27)(25,31)(35,44)(39,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,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=dhi,e^c=ei,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, 6961583);