Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3171 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)(3,4)(7,11)(10,12)(13,14)(15,27)(16,33)(17,23)(18,24)(19,25)(20,26)(21,31)(22,34)(28,41)(29,36)(30,39)(32,38)(35,40)(37,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,28)(17,29)(18,30)(19,31)(20,32)(22,36)(23,37)(24,27)(25,38)(26,39)(33,44)(34,42)(35,43)(40,41);(6,9)(10,12)(13,16,20,17)(14,22,26,40)(15,28,32,29)(18,34,19,35)(21,36,39,41)(23,25,33,24)(27,37,38,44)(30,42,31,43);(6,9)(10,12)(13,17,20,16)(14,23,26,33)(15,29,32,28)(18,34,19,35)(21,37,39,44)(22,25,40,24)(27,36,38,41)(30,42,31,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,34,17,35)(21,27,39,38)(22,23,40,33)(28,42,29,43)(36,37,41,44)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398317);