Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14182 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,7)(4,9)(5,11,19,20)(8,18)(12,28,31,21)(13,23)(14,22)(15,27)(16,25)(17,26)(24,29,32,30);(1,3,5,18,6,13,19,14)(2,7,20,22,10,23,11,8)(4,12,17,29,15,31,16,30)(9,21,25,32,27,28,26,24);(1,3)(2,8)(4,12)(5,18)(6,13)(7,20)(9,24)(10,22)(11,23)(14,19)(15,31)(16,30)(17,29)(21,25)(26,28)(27,32);(1,4,6,15)(2,9)(3,12,13,31)(5,17,19,16)(7,21)(8,24)(10,27)(11,26)(14,30,18,29)(20,25)(22,32)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915 |
| maximal normal subgroup of | #72279915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283767);