Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,11)(14,15)(17,18,23,26)(19,24,29,32)(20,21,25,22)(27,28,30,31);(1,3,5,4)(2,6,8,7)(9,10,13,11)(12,14,16,15)(17,19)(18,24)(20,30)(21,31)(22,28)(23,29)(25,27)(26,32);(1,3,5,4)(2,7,8,6)(9,11,13,10)(12,14,16,15)(17,20)(18,21)(19,27)(22,26)(23,25)(24,28)(29,30)(31,32);(1,3,5,4)(2,7,8,6)(9,12,13,16)(10,14,11,15)(17,21,23,22)(18,25,26,20)(19,28,29,31)(24,30,32,27);(17,22,23,21)(18,25,26,20)(19,28,29,31)(24,27,32,30);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577288);