Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232235 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,12)(8,13,9,11)(16,19,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,32)(31,34)(35,36);(6,11)(7,16)(8,9)(13,14)(17,18)(19,20)(26,32)(33,36);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,20)(14,15)(21,23)(22,26)(24,29)(25,30)(27,33)(28,32)(31,35)(34,36);(5,8,10,9)(6,13,14,11)(7,16,18,17)(12,20,15,19)(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,33,32,36)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240258, #72260903 |
| maximal normal subgroup of | #72240258, #72260903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139444);