Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6370755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#379 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×H(7,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12,15,8)(4,9)(5,7)(6,11)(10,18)(13,25,26,21)(14,22,28,19)(16,23)(17,20)(24,29)(27,31,32,30);(2,7)(5,18)(8,19)(9,20)(10,11)(12,22)(14,28)(17,29)(21,30)(23,24)(25,31)(27,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,19)(9,21)(10,22)(11,12)(16,26)(17,27)(18,28)(20,30)(23,25)(24,31)(29,32);(1,4)(2,9)(3,13)(5,17)(6,16)(7,20)(8,21)(10,24)(11,23)(12,25)(14,27)(15,26)(18,29)(19,30)(22,31)(28,32);(4,16)(9,23)(13,26)(17,29)(20,24)(21,25)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72215901, #72219682, #72219686, … |
| maximal normal subgroup of | #72215901, #72219682, #72219686, … |
| maximal quotient of | C2×#6370755, D4×#1795, #72252480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277951);