Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455952 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16272 |
| 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 | |
1181 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)(3,13)(4,17)(5,12)(6,10)(7,11)(8,15)(9,18)(14,28)(16,24)(19,27)(20,26)(21,25)(22,31)(23,30)(29,32);(1,3,5,20,7,15,6,16)(2,8,10,26,11,13,12,24)(4,14,18,31,19,29,21,30)(9,23,17,32,25,22,27,28);(1,4)(2,9)(3,14,15,29)(5,21)(6,18)(7,19)(8,22,13,23)(10,27)(11,25)(12,17)(16,31,20,30)(24,28,26,32);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,23)(10,17)(11,25)(12,27)(13,22)(15,29)(16,30)(20,31)(24,28)(26,32)Polycyclic presentation:
2^9|b^2=ef,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72287345, #72287348, #72287354, … |
| maximal normal subgroup of | #72287345, #72287348, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363148);