Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394088 |
| 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 | |
3954 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)(10,18)(12,22)(13,27)(14,19)(15,23)(16,20)(17,25)(21,29)(24,30)(26,31)(28,32);(2,7)(3,12)(5,18)(9,20)(10,11)(13,26)(14,15)(16,30)(19,22)(24,25)(28,29)(31,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,19)(9,21)(10,22)(11,23)(12,18)(16,28)(17,29)(20,31)(24,32)(25,27)(26,30);(1,4)(2,9)(3,13)(5,16)(6,17)(7,20)(8,21)(10,24)(11,25)(12,26)(14,28)(15,29)(18,30)(19,31)(22,32)(23,27)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215925, #72216141, #72219687, … |
| maximal normal subgroup of | #72215925, #72216141, #72219687, … |
| maximal quotient of | C2×#394088, #72251220, #72255724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301284);