Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C42 |
| commutator | C4 | C42×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(13,14,16,20,17,22,28,36,18,21,29,35,30,37,43,44)(15,23,25,39,26,32,41,34,24,38,27,42,40,19,31,33);(5,6,7,9,8,10,11,12)(13,15)(14,19,21,32)(16,27)(17,26)(18,24)(20,34,35,33)(22,23,37,38)(25,29)(28,31)(30,40)(36,42,44,39)(41,43);(1,2,3,4)(15,24)(19,32)(23,38)(25,27)(26,40)(31,41)(33,34)(39,42)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,b^a=bg,e^a=ei,c^b=ci,d^b=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72223577, C2×#126524, #72223655, … |
| maximal normal subgroup of | #72223577, C2×#126524, #72223655, … |
| maximal quotient of | C8×#1313, C2×#126524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33720);