Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #42673 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2244 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)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,28,20)(16,21,19,24)(17,30,32,22)(26,38,33,34)(27,35,42,39)(29,37,31,36)(40,43,41,44);(5,7)(6,9)(8,12)(10,11)(15,26)(16,19)(17,29)(20,34)(21,24)(22,36)(25,38)(28,33)(30,37)(31,32)(40,41)(43,44);(1,2,3,4)(13,15,18,28)(14,20,23,25)(16,26,19,33)(17,27,32,42)(21,34,24,38)(22,35,30,39)(29,40,31,41)(36,43,37,44);(6,10)(9,11)(13,16,18,19)(14,21,23,24)(15,26,28,33)(17,31,32,29)(20,34,25,38)(22,37,30,36)(27,41,42,40)(35,44,39,43);(7,12)(9,11)(13,17,18,32)(14,22,23,30)(15,27,28,42)(16,29,19,31)(20,35,25,39)(21,36,24,37)(26,40,33,41)(34,43,38,44)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589531);