Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15258 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2748 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)(3,12)(4,16)(5,19)(6,11)(7,14)(8,18)(9,15)(10,21)(13,27)(17,26)(20,23)(22,28)(24,31)(25,29)(30,32);(1,3,5,14,6,15,21,20)(2,7,10,12,11,23,19,9)(4,13,17,28,18,29,32,31)(8,24,26,25,16,22,30,27);(1,4)(2,8)(3,13)(5,17)(6,18)(7,22)(9,25)(10,26)(11,16)(12,27)(14,28)(15,29)(19,30)(20,31)(21,32)(23,24);(1,3)(2,9)(4,13)(5,20)(6,15)(7,19)(8,25)(10,23)(11,12)(14,21)(16,27)(17,31)(18,29)(22,30)(24,26)(28,32)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72219683, #72219690, … |
| maximal normal subgroup of | #72215953, #72219683, #72219690, … |
| maximal quotient of | C2×#417863, #72251187, #72251236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325059);