Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4052 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)(3,8)(4,9)(5,7)(6,11)(10,17)(12,26)(13,21)(14,23)(15,20)(16,25)(18,19)(22,28)(24,29)(27,31)(30,32);(2,7)(5,17)(8,21)(9,20)(10,11)(13,18)(15,29)(19,23)(22,32)(24,25)(26,31)(27,30);(1,3)(2,8)(4,12)(5,18)(6,14)(7,19)(9,22)(10,21)(11,23)(13,17)(15,30)(16,28)(20,31)(24,32)(25,26)(27,29);(1,4)(2,9)(3,12)(5,15)(6,16)(7,20)(8,22)(10,24)(11,25)(13,27)(14,28)(17,29)(18,30)(19,31)(21,32)(23,26)Polycyclic presentation:
2^9|e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233775, #72233778, C2×#394086, … |
| maximal normal subgroup of | #72233775, #72233778, C2×#394086, … |
| maximal quotient of | #72231778, C2×#394086, #72238108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301282);