Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15368 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1371 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)(4,18)(5,12)(6,10)(7,11)(8,15)(9,20)(14,29)(16,25)(17,23)(19,26)(21,28)(22,27)(24,30)(31,32);(1,3)(2,8)(4,19)(5,17)(6,15)(7,16)(9,26)(10,25)(11,23)(12,13)(14,20)(18,24)(21,31)(22,30)(27,32)(28,29);(8,23)(13,25)(24,29)(26,32);(1,4)(2,9)(3,14,16,30)(5,20)(6,22)(7,21)(8,24,23,29)(10,18)(11,27)(12,28)(13,26,25,32)(15,19,17,31)Polycyclic presentation:
2^9|d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216037, #72216077, … |
| maximal normal subgroup of | #72215958, #72216037, #72216077, … |
| maximal quotient of | C2×#427268, #72255635, #72265157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334464);