Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622875 |
| 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 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3818 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)(2,7)(4,12)(5,14)(6,13)(8,18)(9,11)(10,19)(15,25)(16,26)(17,27)(20,29)(21,23)(22,24)(28,32)(30,31);(4,15)(5,17)(8,20)(12,25)(14,27)(18,29)(21,30)(23,31);(1,4,6,15)(2,8,10,20)(3,12,13,25)(5,16,17,28)(7,18,19,29)(9,21,22,30)(11,23,24,31)(14,26,27,32);(3,13)(7,19)(11,24)(12,25)(14,27)(18,29)(23,31)(26,32)Polycyclic presentation:
2^9|d^2=i,b^a=bf,c^a=cg,h^a=hi,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234840, #72251112, #72251113, … |
| maximal normal subgroup of | #72234840, #72251112, #72251113, … |
| maximal quotient of | #72234743, #72234763, #72256368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530072);