Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 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)(13,14,19,24)(15,20,23,18)(16,25,30,36)(17,26,33,37)(21,31,27,29)(22,32,38,34)(28,35,43,40)(39,42,41,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,23)(14,18,24,20)(16,27,30,21)(17,22,33,38)(25,31,36,29)(26,34,37,32)(28,41,43,39)(35,42,40,44);(13,16,19,30)(14,21,24,27)(15,25,23,36)(17,28,33,43)(18,31,20,29)(22,35,38,40)(26,39,37,41)(32,44,34,42);(2,4)(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,26)(16,28)(18,32)(19,33)(20,34)(21,35)(23,37)(24,38)(25,39)(27,40)(29,42)(30,43)(31,44)(36,41);(6,9)(10,12)(13,18,19,20)(14,23,24,15)(16,29,30,31)(17,32,33,34)(21,25,27,36)(22,37,38,26)(28,42,43,44)(35,39,40,41)Polycyclic presentation:
2^9|a^2=i,b^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,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423983);