Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185261 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,22)(3,12)(4,16)(8,23,25,21)(9,20,11,17)(13,28)(14,15)(18,31)(24,29,26,32)(27,30);(1,2)(3,8)(4,17)(5,9)(6,11)(7,18)(10,19)(12,25)(13,29)(14,24)(15,26)(16,22)(20,31)(21,28)(23,30)(27,32);(4,16)(7,20)(13,30)(17,22)(18,31)(21,32)(23,29)(27,28);(1,3,6,15)(2,8,11,26)(4,13,18,28)(5,14,19,12)(7,21,17,29)(9,24,10,25)(16,30,31,27)(20,32,22,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213205, #72213569, #72213821, … |
| maximal normal subgroup of | #72213205, #72213569, #72213821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92460);