Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4217 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)(3,5)(4,6)(7,8)(15,26)(27,37)(28,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,27)(20,28)(21,26)(22,29)(23,30)(24,31)(25,32)(33,39)(34,37)(35,40)(36,38);(3,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,27)(17,28)(18,26)(22,33)(23,34)(24,35)(25,36)(29,39)(30,37)(31,40)(32,38);(3,7)(5,8)(9,12,14,23)(10,16,18,30)(11,19,21,34)(13,24,25,22)(15,27,26,37)(17,31,32,29)(20,35,36,33)(28,40,38,39);(2,6)(5,8)(9,13,14,25)(10,17,18,32)(11,20,21,36)(12,22,23,24)(15,28,26,38)(16,29,30,31)(19,33,34,35)(27,39,37,40)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302941, #72302998, #72303141, … |
| maximal normal subgroup of | #72302941, #72302998, #72303141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377432);