Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15257 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2423 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)(3,13)(4,17)(5,12)(6,10)(7,11)(8,16)(9,18)(14,23)(15,25)(19,26)(20,22)(21,24)(27,30)(28,32)(29,31);(8,22)(13,25)(14,28)(29,30);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,23)(10,25)(11,22)(12,13)(17,31)(18,30)(19,28)(21,29)(24,32)(26,27);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,23)(10,17)(11,24)(12,26)(13,27)(15,28)(16,29)(20,30)(22,32)(25,31)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216059, #72216166, … |
| maximal normal subgroup of | #72215995, #72216059, #72216166, … |
| maximal quotient of | #72251207, #72251265, C2×#417617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324813);