Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2728 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)(13,14)(15,23)(16,27)(17,19)(18,31)(20,26)(21,28)(22,30)(24,38)(25,36)(29,35)(32,33)(34,44)(37,39)(40,41)(42,43);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,19)(20,33)(21,27)(22,31)(23,36)(34,41)(35,42)(37,38);(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,20,19,33)(16,24,28,39)(18,32,30,26)(21,34,27,41)(22,36,31,23)(29,44,43,40)(35,38,42,37);(7,12)(9,11)(13,16,17,28)(14,21,19,27)(15,24,25,39)(18,29,30,43)(20,34,33,41)(22,35,31,42)(23,37,36,38)(26,40,32,44);(6,10)(9,11)(16,28)(21,27)(24,39)(29,43)(34,41)(35,42)(37,38)(40,44)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,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, 6407716);