Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622880 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,23)(16,24)(17,25)(18,26)(19,22)(20,27)(21,28)(29,31)(30,32);(1,3,7,5)(2,8,12,22)(4,14,17,18)(6,15,20,19)(9,21,25,32)(10,11,13,27)(16,28,29,30)(23,31,26,24);(6,20)(15,19)(16,29)(28,30);(1,4,7,17)(2,9,12,25)(3,14,5,18)(6,16,20,29)(8,21,22,32)(10,26,13,23)(11,24,27,31)(15,28,19,30);(1,5,7,3)(2,10,12,13)(4,14,17,18)(6,19,20,15)(8,11,22,27)(9,23,25,26)(16,28,29,30)(21,31,32,24)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,g^b=ghi,f^c=fh,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236827, #72236893, #72237521, … |
| maximal normal subgroup of | #72237521, #72237528, #72302917, … |
| maximal quotient of | #72303234, #72306342, #72306350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530080);