Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393975 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,21)(10,14)(11,17)(13,24)(16,32)(18,30)(19,31)(20,29)(22,26)(23,28)(25,35)(27,33)(34,36);(5,7)(6,11)(8,16)(9,22)(10,18)(12,27)(13,30)(14,26)(15,21)(17,24)(19,35)(20,34)(23,33)(25,29)(28,36)(31,32);(2,4)(5,8,10,20)(6,12,14,29)(7,16,18,34)(9,23,24,19)(11,25,26,27)(13,31,21,28)(15,32,30,36)(17,35,22,33);(9,24)(13,21)(15,30)(17,22)(19,23)(28,31)(32,36)(33,35)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235458, #72235463, #72249727, … |
| maximal normal subgroup of | #72235458, #72235463, #72249727, … |
| maximal quotient of | #72235747, #72235927, #72239263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301172);