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 | C2×#3271 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2126 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,22)(17,23)(18,20)(19,24)(21,28)(26,38)(27,37)(29,35)(30,36)(31,39)(32,40)(33,34)(41,44)(42,43);(5,6)(7,11)(8,10)(9,12)(14,20)(16,29)(17,18)(21,34)(23,24)(25,37)(26,41)(27,33)(30,42)(31,32)(35,39)(43,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,28)(14,21,24,25)(16,30,32,26)(17,33,18,27)(20,34,23,37)(22,38,40,36)(29,42,31,41)(35,44,39,43);(7,12)(9,11)(13,16,19,32)(14,22,24,40)(15,26,28,30)(17,31,18,29)(20,35,23,39)(21,36,25,38)(27,41,33,42)(34,43,37,44);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,27,28,33)(16,31,32,29)(21,34,25,37)(22,39,40,35)(26,41,30,42)(36,43,38,44)Polycyclic presentation:
2^9|c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962078);