Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,26,20)(16,28,30,21)(17,22,32,40)(19,24,33,41)(27,42,34,37)(29,38,36,44)(31,43,35,39);(1,2,4,3)(6,9)(10,12)(13,15)(14,20)(16,29)(17,31)(18,26)(19,34)(21,38)(22,39)(23,25)(24,42)(27,33)(28,44)(30,36)(32,35)(37,41)(40,43);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,30)(14,21,23,28)(15,17,26,32)(19,35,33,31)(20,22,25,40)(24,43,41,39)(27,36,34,29)(37,44,42,38);(13,17,18,32)(14,22,23,40)(15,16,26,30)(19,36,33,29)(20,21,25,28)(24,44,41,38)(27,35,34,31)(37,43,42,39);(15,26)(16,30)(20,25)(21,28)(27,34)(31,35)(37,42)(39,43)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347901);