Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3188 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)(15,19)(16,21)(17,20)(18,23)(22,30)(24,32)(25,31)(26,34)(27,36)(28,37)(29,35)(33,40)(38,42)(39,43)(41,44);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,23,34)(16,27,28,24)(17,25,30,40)(20,31,22,33)(21,36,37,32)(29,41,42,39)(35,44,38,43);(5,7)(6,9)(8,12)(10,11)(14,20)(17,30)(19,31)(21,35)(22,23)(25,40)(29,42)(32,43)(33,34)(36,44)(37,38)(39,41);(7,12)(9,11)(13,16,18,28)(14,21,23,37)(15,24,26,27)(17,29,30,42)(19,32,34,36)(20,35,22,38)(25,39,40,41)(31,43,33,44);(6,10)(9,11)(16,28)(21,37)(24,27)(29,42)(32,36)(35,38)(39,41)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6407365);