Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,15,9,23)(7,16,19,13,17,31,24,32)(8,12,21,28,22,29,25,27)(18,34,36,26,20,33,35,30);(1,2,3,4)(6,12)(7,17)(13,30)(14,28)(15,29)(16,33)(18,20)(19,24)(23,27)(26,32)(31,34)(35,36);(5,7,10,19,11,17,9,24)(6,13,14,31,15,32,23,16)(8,20,21,35,22,18,25,36)(12,26,28,33,29,30,27,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312115, #72312123, #72312162, … |
| maximal normal subgroup of | #72312115, #72312123, #72312162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54538);