Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 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)(13,14,20,26)(15,27,32,38)(16,22,17,23)(18,25,19,24)(21,30,39,31)(28,43,33,37)(29,40,44,36)(34,42,35,41);(5,6)(7,11)(8,10)(9,12)(13,15,20,32)(14,21,26,39)(16,33,17,28)(18,31,19,30)(22,40,23,36)(24,27,25,38)(29,34,44,35)(37,41,43,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,28)(17,20)(18,35)(19,34)(21,36)(23,26)(24,42)(25,41)(27,43)(29,30)(31,44)(32,33)(37,38)(39,40);(7,12)(9,11)(13,17,20,16)(14,23,26,22)(15,29,32,44)(18,35,19,34)(21,37,39,43)(24,42,25,41)(27,40,38,36)(28,31,33,30);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,34,17,35)(21,38,39,27)(22,41,23,42)(28,44,33,29)(36,43,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901170);