Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14,20,26)(15,21,29,23)(16,30,35,40)(17,36,28,38)(18,24,19,25)(22,34,41,33)(27,39,43,44)(31,37,32,42);(1,2)(3,4)(6,10)(11,12)(13,15,20,29)(14,21,26,23)(16,31,35,32)(17,18,28,19)(22,39,41,44)(24,38,25,36)(27,34,43,33)(30,37,40,42);(5,7,9,8)(6,11,10,12)(13,16,18,33,20,35,19,34)(14,22,25,30,26,41,24,40)(15,27,17,31,29,43,28,32)(21,37,38,39,23,42,36,44);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,15,20,28,18,29)(14,23,24,38,26,21,25,36)(16,32,34,43,35,31,33,27)(22,37,40,44,41,42,30,39);(6,10)(11,12)(13,18,20,19)(14,24,26,25)(15,28,29,17)(16,33,35,34)(21,38,23,36)(22,40,41,30)(27,32,43,31)(37,39,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469214);