Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2752 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16660 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1170 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)(3,11)(4,15)(5,19)(6,10)(7,14)(8,18)(9,21)(12,28)(13,27)(16,32)(17,25)(20,26)(22,30)(23,31)(24,29);(7,22)(11,27)(23,24)(28,32);(1,3,5,14,6,13,21,30)(2,7,9,11,10,22,19,27)(4,16,17,31,18,12,20,29)(8,24,25,28,15,23,26,32);(1,4)(2,8,10,15)(3,12)(5,20)(6,18)(7,23,22,24)(9,26,19,25)(11,28,27,32)(13,16)(14,31)(17,21)(29,30)Polycyclic presentation:
2^9|c^2=f,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251849, #72251854, #72251857, … |
| maximal normal subgroup of | #72251849, #72251854, #72251857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376877);