Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471888 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16808 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1177 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)(3,7,13,20)(4,15)(5,9)(6,18)(8,16)(10,11)(12,26,27,32)(14,29,25,22)(17,24)(19,23)(21,30,31,28);(1,3,11,25,5,13,6,14)(2,7,18,29,9,20,10,22)(4,12,16,28,17,27,19,30)(8,21,15,26,23,31,24,32);(6,11)(7,20)(8,23)(10,18)(12,27)(14,25)(16,19)(21,31);(1,4)(2,8)(3,12)(5,17)(6,19)(7,21)(9,23)(10,24)(11,16)(13,27)(14,30)(15,18)(20,31)(22,32)(25,28)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=egh,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72286957, #72287326, #72287354, … |
| maximal normal subgroup of | #72286957, #72287326, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379084);