Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ 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:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,23)(15,19,25,33)(16,21,28,37)(17,20,30,22)(24,32,27,36)(26,31,40,34)(29,35,42,38)(39,43,41,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,27)(17,26)(18,25)(20,31)(21,36)(22,34)(23,33)(24,28)(29,41)(30,40)(32,37)(35,44)(38,43)(39,42);(1,3,4,2)(7,11)(9,12)(14,20)(17,30)(19,31)(21,35)(22,23)(26,40)(29,42)(32,43)(33,34)(36,44)(37,38)(39,41);(7,11)(9,12)(13,16)(14,21)(15,24)(17,29)(18,28)(19,32)(20,35)(22,38)(23,37)(25,27)(26,39)(30,42)(31,43)(33,36)(34,44)(40,41);(6,10)(9,12)(15,25)(16,28)(19,33)(21,37)(26,40)(29,42)(31,34)(35,38)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377991);