Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,17,29)(4,18,13,10)(5,11)(6,9)(7,15)(8,19,26,21)(14,24)(16,30,20,23)(22,27)(25,32)(28,31);(1,3,13,15,6,17,4,14)(2,7,18,29,9,24,10,12)(5,20,21,31,22,16,19,32)(8,23,11,28,26,30,27,25);(1,4,6,13)(2,8,9,26)(3,14,17,15)(5,21,22,19)(7,23,24,30)(10,27,18,11)(12,25,29,28)(16,32,20,31);(2,9)(7,24)(8,26)(10,18)(11,27)(12,29)(23,30)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237532, #72237542, #72237544, … |
| maximal normal subgroup of | #72237532, #72237542, #72237544, … |
| maximal quotient of | #72312254, #72312563, C2×#494750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401946);