Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16476 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 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)(4,10,19,27)(5,8)(6,11)(7,12)(9,16)(14,22)(15,24,30,29)(17,32,31,25)(18,23,21,26)(20,28);(2,8)(3,14)(5,6)(9,13)(10,23)(11,12)(15,17)(16,20)(18,21)(24,29)(26,27)(30,31);(1,3,5,20,7,16,6,14)(2,9,11,28,12,13,8,22)(4,17,18,15,19,31,21,30)(10,25,26,24,27,32,23,29);(1,4)(2,10)(3,15,16,30)(5,18)(6,21)(7,19)(8,23)(9,24,13,29)(11,26)(12,27)(14,17,20,31)(22,25,28,32)Polycyclic presentation:
2^9|a^2=efg,c^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251693, #72251699, … |
| maximal normal subgroup of | #72251689, #72251693, #72251699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370736);