Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394341 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2622 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)(3,11)(4,8)(5,9)(6,14)(7,10)(12,16)(13,25)(15,28)(17,23)(18,27)(19,24)(20,22)(21,30)(26,31)(29,32);(1,3,11,2,7,14,6,10)(4,12,20,23,17,22,16,8)(5,13,25,9,19,27,18,24)(15,26,31,28,30,32,29,21);(1,4)(2,8)(3,12)(5,15)(6,20)(7,17)(9,21)(10,23)(11,16)(13,26)(14,22)(18,31)(19,30)(24,28)(25,29)(27,32);(1,5)(2,9)(3,13)(4,15)(6,18)(7,19)(8,21)(10,24)(11,25)(12,26)(14,27)(16,29)(17,30)(20,31)(22,32)(23,28)Polycyclic presentation:
2^9|b^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72219687, #72219691, … |
| maximal normal subgroup of | #72215953, #72219687, #72219691, … |
| maximal quotient of | #72251234, C2×#394342, #72312642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301538);