Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2178 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)(8,12)(9,10)(13,14,19,25)(15,22,29,21)(16,30,28,38)(17,32,35,41)(18,26,20,24)(23,34,42,36)(27,43,44,40)(31,37,33,39);(1,2)(3,4)(6,10)(11,12)(13,15)(14,21)(16,20)(17,33)(18,28)(19,29)(22,25)(23,40)(24,38)(26,30)(27,36)(31,35)(32,37)(34,44)(39,41)(42,43);(5,7,9,8)(6,11,10,12)(13,16,20,15,19,28,18,29)(14,22,26,30,25,21,24,38)(17,33,36,44,35,31,34,27)(23,37,32,40,42,39,41,43);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,34,19,35,20,36)(14,23,24,41,25,42,26,32)(15,27,16,31,29,44,28,33)(21,37,30,43,22,39,38,40);(6,10)(11,12)(13,18,19,20)(14,24,25,26)(15,28,29,16)(17,34,35,36)(21,38,22,30)(23,41,42,32)(27,33,44,31)(37,40,39,43)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469496);