Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,22)(15,25,28,35)(16,21,31,40)(17,24,18,23)(20,29,36,27)(26,37,42,44)(30,34,41,43)(32,39,33,38);(1,2,3,4)(13,15,18,29,19,28,17,27)(14,20,24,35,22,36,23,25)(16,30,33,42,31,41,32,26)(21,37,39,43,40,44,38,34);(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,26)(17,33)(18,32)(19,31)(20,34)(22,40)(23,39)(24,38)(25,37)(27,30)(28,42)(29,41)(35,44)(36,43);(5,8,9,7)(6,10,11,12)(15,27)(17,18)(20,25)(23,24)(26,30)(28,29)(32,33)(34,37)(35,36)(38,39)(41,42)(43,44);(14,22)(15,28)(16,31)(23,24)(27,29)(32,33)(34,43)(37,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359407);