Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2686 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15274 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1354 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)(3,13,17,25)(4,18,22,20)(5,7)(6,11)(8,12)(9,15,26,16)(10,21,28,23)(14,24)(19,27)(29,32)(30,31);(1,3)(2,9)(4,19)(5,13)(6,16)(7,15)(8,17)(10,27)(11,25)(12,26)(14,21)(18,24)(20,31)(22,29)(23,30)(28,32);(1,4)(2,10)(3,14)(5,20)(6,21)(7,23)(8,22)(9,24)(11,18)(12,28)(13,27)(15,19)(16,29)(17,30)(25,32)(26,31);(1,5,6,2,8,11,7,12)(3,9,15,13,17,26,16,25)(4,20,21,10,22,18,23,28)(14,24,19,27,30,31,29,32)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72216144, #72219482, … |
| maximal normal subgroup of | #72215906, #72216144, #72219482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327261);