Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23573 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(18,22)(20,23)(24,29)(25,30)(26,32)(28,31);(9,10)(11,14)(12,13)(15,16)(17,18,23,30,21,25,20,22)(19,26,31,29,27,24,28,32);(2,6)(7,8)(10,13)(14,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,31)(25,26)(30,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,27)(24,26)(28,31)(29,32);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(18,25)(22,30)(24,26)(29,32)Polycyclic presentation:
2^9|b^2=fi,d^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251258, #72256852, #72256928 |
| maximal normal subgroup of | #72251258, #72256852, #72256928 |
| maximal quotient of | #72308471 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6444071);