Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15,20,30)(12,22,23,34)(13,18)(14,17)(16,24,31,26)(19,35)(21,29,25,33)(27,32)(28,38)(36,39)(37,40);(9,11)(10,15,17,30)(12,19)(13,25)(14,20)(16,28,31,40)(18,33,32,29)(21,27)(22,39,34,35)(23,36)(24,38)(26,37);(2,5)(6,8)(12,23)(16,31)(19,36)(22,34)(24,26)(28,40)(35,39)(37,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,31)(18,34)(20,36)(21,38)(22,32)(24,27)(25,37)(29,39)(30,40)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317836, #72318275, #72318313, … |
| maximal normal subgroup of | #72317836, #72318275, #72318313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300756);