Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,26)(19,23,28,32)(20,25,21,24)(27,30,29,31);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(10,14)(13,15)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32);(2,5)(6,8)(10,14)(13,15)(17,20,22,21)(18,25,26,24)(19,27,28,29)(23,30,32,31)Polycyclic presentation:
2^9|b^2=i,c^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72256825, #72256840, #72256982, … |
| maximal normal subgroup of | #72256825, #72256840, #72256982, … |
| maximal quotient of | #72307882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7438896);