Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623308 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,26)(14,17)(18,23)(19,24)(20,28)(21,27)(22,36)(25,31)(29,35)(30,34)(32,37)(33,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,28)(18,29)(19,30)(23,35)(24,34)(25,33)(26,36)(31,38)(32,39)(37,40);(3,8)(5,7)(10,16)(12,23)(13,24)(15,27)(17,31)(18,19)(21,35)(22,34)(25,37)(26,32)(28,38)(29,30)(33,40)(36,39);(2,6)(5,7)(11,20)(15,28)(21,33)(22,34)(27,38)(29,39)(30,36)(35,40);(10,17)(15,28)(16,31)(18,32)(19,26)(27,38)(29,39)(30,36)Polycyclic presentation:
2^9|g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=di,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265451, #72265463, #72265516, … |
| maximal normal subgroup of | #72265451, #72265463, #72265516, … |
| maximal quotient of | #72320553, C2×#7623308, #72343736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530504);