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 | C2×#3285 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,29,20)(16,30,33,40)(17,24,19,22)(21,34,38,32)(26,42,44,37)(27,39,28,36)(31,41,43,35);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,31)(17,28)(18,29)(19,27)(21,37)(22,36)(23,25)(24,39)(26,32)(30,35)(33,43)(34,44)(38,42)(40,41);(13,16,18,33)(14,21,23,38)(15,26,29,44)(17,32,19,34)(20,35,25,41)(22,30,24,40)(27,31,28,43)(36,42,39,37);(2,4)(5,7,8,11)(6,10,9,12)(14,22)(15,27)(17,19)(21,30)(23,24)(26,31)(28,29)(32,34)(36,39)(37,42)(38,40)(43,44);(6,9)(10,12)(13,17,18,19)(14,22,23,24)(15,28,29,27)(16,32,33,34)(20,36,25,39)(21,30,38,40)(26,43,44,31)(35,42,41,37)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fi,e^2=i,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=cg,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 7423815);