Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(10,12)(13,14,16,18)(15,17,19,20)(21,22,26,25)(23,28,24,27);(2,4)(9,10)(11,12)(15,19)(17,20)(21,23,26,24)(22,27,25,28);(5,6,8,7)(9,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,23)(22,28,25,27);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,15,16,19)(14,17,18,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^9|a^2=fh,b^2=eg,c^2=egh,d^2=fg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| derived subgroup of | #72266606 |
| Fitting subgroup of | #72266606 |
| maximal subgroup of | #72266606, #72267034, #72267037, … |
| maximal normal subgroup of | #72266606, #72267034, #72267037, … |
| maximal quotient of | #72266611 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67550);