Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14,20,26)(15,21,30,39)(16,31,17,23)(18,25,19,24)(22,34,40,32)(27,36,33,41)(28,38,29,37)(35,43,44,42);(2,3)(6,9)(7,11)(13,15)(14,21)(16,27)(17,33)(18,28)(19,29)(20,30)(22,35)(23,41)(24,37)(25,38)(26,39)(31,36)(32,42)(34,43)(40,44);(5,6)(7,10)(8,9)(11,12)(13,16,20,17)(14,22,26,40)(15,27,30,33)(18,32,19,34)(21,35,39,44)(23,24,31,25)(28,42,29,43)(36,38,41,37);(1,3,4,2)(6,9)(10,12)(13,17,20,16)(14,23,26,31)(15,27,30,33)(18,32,19,34)(21,36,39,41)(22,24,40,25)(28,43,29,42)(35,38,44,37);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,28,30,29)(16,32,17,34)(21,37,39,38)(22,31,40,23)(27,42,33,43)(35,36,44,41)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053218);