Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,32)(17,23)(18,36)(19,25)(20,24)(21,30)(22,34)(27,37)(28,43)(29,39)(31,42)(33,38)(35,40)(41,44);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,33)(17,27)(18,31)(19,30)(20,29)(22,38)(23,41)(24,42)(25,26)(28,34)(32,43)(35,44)(36,39)(37,40);(1,2,4,3)(6,10)(9,12)(13,16)(14,22)(15,27)(17,18)(19,34)(20,35)(21,37)(23,24)(25,32)(26,41)(28,31)(29,33)(30,44)(36,40)(38,42)(39,43);(1,3,4,2)(13,17,19,35)(14,23,25,40)(15,28,30,33)(16,18,34,20)(21,38,26,43)(22,24,32,36)(27,31,44,29)(37,42,41,39);(13,18,19,20)(14,24,25,36)(15,29,30,31)(16,17,34,35)(21,39,26,42)(22,23,32,40)(27,33,44,28)(37,43,41,38)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6348851);