Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16512 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(5,6)(9,23)(10,26)(11,12)(15,29)(16,17)(18,30)(19,20)(22,28)(24,31);(1,2,6,11,7,12,5,8)(3,14,13,23,17,25,16,9)(4,10,19,26,21,28,20,22)(15,27,18,31,30,32,29,24);(1,3,5,13,7,17,6,16)(2,9,8,14,12,23,11,25)(4,18,20,30,21,29,19,15)(10,27,22,31,28,32,26,24);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,22)(9,24)(11,26)(12,28)(13,18)(14,27)(16,29)(17,30)(23,31)(25,32)Polycyclic presentation:
2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251790, #72251801, … |
| maximal normal subgroup of | #72251783, #72251790, #72251801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372164);