Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,19,26,35)(16,28,30,40)(17,21,33,22)(20,32,38,31)(24,41,43,37)(25,36,27,39)(29,44,42,34);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,29)(17,25)(18,26)(20,37)(21,36)(22,39)(23,35)(24,32)(27,33)(28,44)(30,42)(31,43)(34,40)(38,41);(13,16,18,30)(14,20,23,38)(15,24,26,43)(17,32,33,31)(19,34,35,44)(21,28,22,40)(25,42,27,29)(36,37,39,41);(2,4)(5,7,9,8)(6,11,12,10)(14,21)(15,25)(17,33)(20,28)(22,23)(24,42)(26,27)(29,43)(31,32)(36,39)(37,41)(38,40);(5,8,9,7)(6,11,12,10)(15,26)(16,30)(19,35)(20,38)(25,27)(28,40)(31,32)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429887);