Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,22,17,21,15,19)(16,24,26,27,25,20,23,28);(5,7,10,6)(8,11,9,12)(13,14,15,19,17,21,18,22)(16,20,23,27,25,24,26,28);(1,2,3,4)(5,8,10,9)(6,11,7,12)(13,15,17,18)(14,19,21,22)(16,23,25,26)(20,27,24,28);(5,9,10,8)(6,11,7,12)(13,16,17,25)(14,20,21,24)(15,23,18,26)(19,27,22,28);(16,25)(20,24)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239412, #72249595, #72249598 |
| maximal normal subgroup of | #72239412, #72249595, #72249598 |
| maximal quotient of | #72239348, #72239363, #72239393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283670);