Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,17,19,24,18,23)(15,25,35,21,28,40,27,37)(16,22,31,34,33,39,32,30)(26,41,42,36,29,44,43,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,24,37)(16,29,33,26)(17,25,23,40)(18,35,20,27)(22,38,39,36)(30,41,34,44)(31,42,32,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,34)(18,32)(19,33)(20,31)(21,36)(23,30)(24,39)(25,41)(27,43)(28,29)(35,42)(37,38)(40,44);(7,12)(9,11)(13,17,20,24,19,23,18,14)(15,21,35,40,28,37,27,25)(16,30,31,22,33,34,32,39)(26,38,42,41,29,36,43,44);(6,10)(9,11)(13,18,19,20)(14,23,24,17)(15,27,28,35)(16,31,33,32)(21,25,37,40)(22,34,39,30)(26,42,29,43)(36,44,38,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914350);