Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #42806 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,15,24,18,23,19,20)(16,27,28,21,30,35,25,38)(17,31,26,36,34,22,32,39)(29,44,42,40,33,43,41,37);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,23,24)(16,28,30,25)(17,32,34,26)(21,27,38,35)(22,39,31,36)(29,41,33,42)(37,44,40,43);(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,21,23,38)(15,25,19,28)(17,33,34,29)(20,35,24,27)(22,40,31,37)(26,42,32,41)(36,43,39,44);(1,3,4,2)(7,12)(9,11)(13,17,18,34)(14,22,23,31)(15,26,19,32)(16,29,30,33)(20,36,24,39)(21,37,38,40)(25,41,28,42)(27,43,35,44);(6,10)(9,11)(14,23)(17,34)(20,24)(21,38)(26,32)(27,35)(29,33)(41,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913727);