Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #24438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,24,30,31)(21,25,22,26)(27,28,32,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,30)(18,24,20,31)(21,29,22,28)(25,27,26,32);(10,14)(13,16)(17,20,23,18)(19,27,30,32)(21,25,22,26)(24,28,31,29);(19,28)(21,22)(24,32)(25,26)(27,31)(29,30);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,25,20,26)(19,29,30,28)(24,27,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257411, #72257502, #72264023, … |
| maximal normal subgroup of | #72257411, #72257502, #72264023, … |
| maximal quotient of | #72308301 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349306);