Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418482 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1725 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)(4,9)(5,8)(6,22)(7,12)(10,15)(11,20)(14,26)(16,25)(17,29)(18,27)(19,28)(21,24)(23,32)(30,31);(1,3)(2,8)(4,14)(5,20)(6,21)(7,15)(9,25)(10,11)(12,24)(13,22)(16,18)(17,32)(19,31)(23,28)(26,29)(27,30);(1,4)(2,9)(3,14)(5,16)(6,17)(7,19)(8,23)(10,26)(11,27)(12,28)(13,30)(15,31)(18,20)(21,32)(22,29)(24,25);(1,5,20,3,7,21,6,15)(2,10,22,24,12,13,11,8)(4,16,18,14,19,32,17,31)(9,26,29,25,28,30,27,23)Polycyclic presentation:
2^9|d^2=ei,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^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#418483, #72237690, #72240435, … |
| maximal normal subgroup of | C2×#418483, #72237690, #72240435, … |
| maximal quotient of | C2×#418483, #72251319, #72255899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325679);