Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622886 |
| 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×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3823 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,23)(13,24)(15,20)(16,21)(17,22)(18,26)(19,27)(25,31)(28,30)(29,32);(1,3,6,12)(2,7,10,18)(4,13,15,25)(5,14,17,26)(8,19,20,29)(9,11,22,23)(16,27,28,32)(21,24,30,31);(3,12)(5,17)(7,18)(11,23)(13,25)(16,28)(19,29)(24,31);(1,4)(2,8)(3,13)(5,16)(6,15)(7,19)(9,21)(10,20)(11,24)(12,25)(14,27)(17,28)(18,29)(22,30)(23,31)(26,32);(4,15)(8,20)(13,25)(16,28)(19,29)(21,30)(24,31)(27,32)Polycyclic presentation:
2^9|b^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ci,g^b=ghi,f^c=fh,e^d=ei,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233206, #72233535, #72234103, … |
| maximal normal subgroup of | #72233206, #72233535, #72234103, … |
| maximal quotient of | #72233167, #72233302, #72233533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530083);