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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 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)(13,14,19,32,18,24,17,23)(15,20,26,35,28,37,27,36)(16,22,29,21,31,38,30,39)(25,34,42,33,41,44,40,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,28)(14,20,24,37)(16,25,31,41)(17,27,19,26)(21,33,39,43)(22,34,38,44)(23,36,32,35)(29,42,30,40);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,33)(22,32)(23,38)(24,39)(26,40)(27,42)(28,41)(34,35)(36,44)(37,43);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,25)(17,30)(18,31)(19,29)(20,34)(21,32)(23,39)(24,38)(26,42)(27,40)(28,41)(33,35)(36,43)(37,44);(6,10)(9,11)(13,17,18,19)(14,23,24,32)(15,26,28,27)(16,29,31,30)(20,35,37,36)(21,38,39,22)(25,40,41,42)(33,34,43,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,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, 6913484);