Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14885 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2621 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)(4,15)(5,18)(6,10)(7,13)(8,17)(9,20)(12,26)(14,24)(16,23)(19,21)(22,31)(25,28)(27,29)(30,32);(1,3)(2,7)(4,12)(5,19)(6,14)(8,22)(9,24)(10,21)(11,18)(13,20)(15,29)(16,31)(17,28)(23,26)(25,30)(27,32);(7,21)(11,24)(22,29)(25,26);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,22,21,29)(9,23,18,30)(11,25,24,26)(13,27)(14,28)(19,31)(20,32)Polycyclic presentation:
2^9|d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216029, #72216037, … |
| maximal normal subgroup of | #72215972, #72216029, #72216037, … |
| maximal quotient of | #72237670, #72239427, #72239429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311188);