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 | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,18,32)(16,26,28,34)(17,22,30,39)(21,33,37,24)(25,35,31,44)(27,42,29,43)(36,41,38,40);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,23,32)(16,24,28,33)(17,25,30,31)(21,34,37,26)(22,35,39,44)(27,40,29,41)(36,43,38,42);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,24)(17,29)(18,33)(19,28)(20,34)(22,38)(23,37)(25,41)(26,32)(27,30)(31,40)(35,42)(36,39)(43,44);(2,4)(7,12)(9,11)(13,17)(14,22)(15,25)(16,27)(18,31)(19,30)(20,35)(21,36)(23,39)(24,40)(26,42)(28,29)(32,44)(33,41)(34,43)(37,38);(6,10)(9,11)(16,28)(17,30)(21,37)(22,39)(24,33)(25,31)(26,34)(35,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,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, 6961455);