Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,14)(5,17)(7,20)(8,23)(10,18)(11,25)(13,27)(15,24)(16,19)(21,31)(26,29)(30,32);(1,2)(3,7)(4,15)(5,18)(6,9)(8,16)(10,17)(11,21)(12,20)(13,28)(14,23)(19,24)(22,27)(25,31)(26,30)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,21)(9,20)(10,22)(14,25)(15,30)(16,26)(17,27)(18,28)(19,29)(23,32)(24,31);(1,4)(2,8,9,15)(3,11)(5,19)(6,16)(7,21,20,30)(10,24,18,23)(12,26)(13,29)(14,17)(22,31,28,32)(25,27)Polycyclic presentation:
2^9|d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72240432, #72240435, #72240458, … |
| maximal normal subgroup of | #72240432, #72240435, #72240458, … |
| maximal quotient of | #72251293, #72251311, C2×#427262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334458);