Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13877 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,6,14,7,13)(3,9,17,28,16,29,18,27)(4,19,15,26,21,32,22,10)(5,11,20,25,23,30,24,31);(1,3)(2,9)(4,20)(5,15)(6,16)(7,17)(8,18)(10,30)(11,26)(12,27)(13,28)(14,29)(19,25)(21,24)(22,23)(31,32);(1,4,7,22,6,21,8,15)(2,10,12,19,14,26,13,32)(3,5,17,20,16,23,18,24)(9,25,27,11,29,31,28,30);(1,5)(2,11)(3,15)(4,17)(6,23)(7,20)(8,24)(9,26)(10,29)(12,31)(13,25)(14,30)(16,22)(18,21)(19,28)(27,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72219686, #72219688, … |
| maximal normal subgroup of | #72215931, #72219686, #72219688, … |
| maximal quotient of | C2×#360561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267757);