Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2557 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15270 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1393 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)(3,11,13,7)(4,8,16,25)(5,18)(6,9)(10,21)(12,22)(14,23,19,27)(15,32)(17,28,20,26)(24,31)(29,30);(1,3)(2,7)(4,15)(5,19)(6,13)(8,24)(9,11)(10,27)(12,17)(14,21)(16,31)(18,23)(20,30)(22,26)(25,32)(28,29);(1,4)(2,8,9,25)(3,12)(5,20)(6,16)(7,22,11,29)(10,28,18,26)(13,30)(14,31)(15,19)(17,21)(23,32,27,24);(7,11)(22,29)(23,27)(24,32)Polycyclic presentation:
2^9|a^2=ef,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216029, #72216130, … |
| maximal normal subgroup of | #72215936, #72216029, #72216130, … |
| maximal quotient of | #72237668, #72237714, C2×#419463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326659);