Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3044 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)(13,14,18,24,20,26,19,25)(15,23,29,37,17,35,28,21)(16,30,33,22,34,39,32,40)(27,41,42,38,31,36,43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,17)(14,21,26,37)(16,27,34,31)(18,28,19,29)(22,36,40,41)(23,24,35,25)(30,44,39,38)(32,42,33,43);(6,9)(7,11)(13,16)(14,22)(15,27)(17,31)(18,32)(19,33)(20,34)(21,36)(23,38)(24,30)(25,39)(26,40)(28,42)(29,43)(35,44)(37,41);(5,7,8,11)(6,10,9,12)(13,17,20,15)(14,23,26,35)(16,31,34,27)(18,28,19,29)(21,24,37,25)(22,38,40,44)(30,41,39,36)(32,42,33,43);(6,9)(10,12)(13,18,20,19)(14,24,26,25)(15,28,17,29)(16,32,34,33)(21,35,37,23)(22,30,40,39)(27,42,31,43)(36,44,41,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 6420579);