Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394483 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14725 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2141 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)(3,7)(4,15,16,26)(5,10)(6,9)(8,17,23,19)(11,13)(12,28,29,32)(14,22)(18,25)(20,24)(21,30,27,31);(1,2)(3,11,13,7)(4,8)(5,10)(6,9)(12,27,29,21)(14,25,18,22)(15,17)(16,23)(19,26)(20,24)(28,30,32,31);(1,3)(2,7)(4,12)(5,18)(6,13)(8,21)(9,11)(10,25)(14,20)(15,32)(16,29)(17,31)(19,30)(22,24)(23,27)(26,28);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,23)(10,26)(11,27)(13,29)(14,31)(15,24)(17,20)(18,30)(22,32)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319976, #72320000, #72320059, … |
| maximal normal subgroup of | #72319976, #72320000, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302815);