Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25124 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,22)(19,26)(24,29)(27,32);(10,14)(11,15)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(11,15)(13,16);(1,3,4,7)(2,6,5,8)(19,26)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|c^2=ghi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239598, #72251242, #72256827, … |
| maximal normal subgroup of | #72239598, #72251242, #72256827, … |
| maximal quotient of | #72239662, #72239733, #72308282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368153);