Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2104 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(20,22)(24,31)(27,28)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,24,28,30)(20,29,22,25)(26,32,27,31);(10,14)(11,15)(17,19,22,26,21,28,20,27)(18,24,25,32,23,30,29,31);(2,6)(5,7)(17,20,21,22)(18,25,23,29)(19,27,28,26)(24,32,30,31);(3,8)(5,7)(11,15)(13,16)(18,23)(24,30)(25,29)(31,32)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281234, #72281357, #72319809 |
| maximal normal subgroup of | #72281234, #72281357, #72319809 |
| maximal quotient of | #72320432 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669130);