Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C24 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,9,11,8,10,12,7)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(6,10)(7,11)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(5,7,12,10,8,11,9,6)(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=h,g^2=i,b^a=be,d^a=di,c^b=chi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72318348, #72318433, #72319519, … |
| maximal normal subgroup of | #72318348, #72318433, #72319519, … |
| maximal quotient of | #72319047, #72321583, C2×#183309 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90505);