Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45245 | 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)(13,14,19,23,20,24,18,17)(15,25,28,38,29,40,27,21)(16,30,33,35,34,22,36,32)(26,42,43,37,31,41,44,39);(5,6)(7,11)(8,10)(9,12)(13,15,20,29)(14,21,24,38)(16,31,34,26)(17,25,23,40)(18,28,19,27)(22,39,30,37)(32,41,35,42)(33,43,36,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,35)(18,36)(19,33)(20,34)(21,37)(23,32)(24,30)(25,41)(27,44)(28,43)(29,31)(38,39)(40,42);(7,12)(9,11)(13,17,18,24,20,23,19,14)(15,21,27,40,29,38,28,25)(16,32,36,22,34,35,33,30)(26,39,44,41,31,37,43,42);(6,10)(9,11)(13,18,20,19)(14,23,24,17)(15,27,29,28)(16,33,34,36)(21,25,38,40)(22,35,30,32)(26,43,31,44)(37,42,39,41)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,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913998);