Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #36085 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2674 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)(8,10)(9,12)(13,14,20,25)(15,21,29,42)(16,30,27,44)(17,33,26,43)(18,24,19,36)(22,34,39,32)(23,31,38,35)(28,40,37,41);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,31)(17,34)(18,28)(19,37)(20,29)(22,43)(23,44)(24,40)(25,42)(26,32)(27,35)(30,38)(33,39)(36,41);(13,16)(14,22)(15,26)(17,29)(18,32)(19,34)(20,27)(21,38)(23,42)(24,44)(25,39)(28,35)(30,36)(31,37)(33,41)(40,43);(2,4)(13,17,20,26)(14,23,25,38)(15,27,29,16)(18,35,19,31)(21,39,42,22)(24,43,36,33)(28,32,37,34)(30,40,44,41);(6,10)(9,11)(13,18,20,19)(14,24,25,36)(15,28,29,37)(16,32,27,34)(17,31,26,35)(21,40,42,41)(22,44,39,30)(23,33,38,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,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,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421867);