Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1698 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)(3,9)(4,17)(5,8)(6,11)(7,12)(10,20)(13,23)(14,28)(15,21)(16,24)(18,26)(19,25)(22,29)(27,32)(30,31);(2,8)(3,13)(4,18)(5,6)(11,12)(15,16)(17,25)(19,20)(21,23)(22,28)(27,29)(31,32);(1,3)(2,9)(4,14)(5,15)(6,13)(7,16)(8,21)(10,22)(11,23)(12,24)(17,31)(18,29)(19,30)(20,27)(25,28)(26,32);(1,4,5,18,7,19,6,20)(2,10,11,25,12,26,8,17)(3,14,15,29,16,30,13,27)(9,22,23,28,24,32,21,31)Polycyclic presentation:
2^9|d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216077, #72216192, … |
| maximal normal subgroup of | #72215995, #72216077, #72216192, … |
| maximal quotient of | #72237181, C2×#418471, #72252146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325667);