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 | |
4075 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)(13,23)(16,32)(18,29)(19,30)(20,28)(22,25)(24,27)(26,33)(31,34)(35,36);(5,7)(6,11)(8,16)(9,22)(10,18)(12,26)(13,29)(14,25)(15,21)(17,23)(19,35)(20,34)(24,33)(27,31)(28,36)(30,32);(2,4)(9,23)(11,25)(13,21)(17,22)(19,24)(26,36)(27,30)(33,35);(5,8,10,20)(6,12,14,28)(7,16,18,34)(9,24,23,19)(11,26,25,36)(13,30,21,27)(15,31,29,32)(17,35,22,33)Polycyclic presentation:
2^9|d^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 | #72235198, #72249727, #72251694, … |
| maximal normal subgroup of | #72235198, #72249727, #72251694, … |
| maximal quotient of | #72235992, #72302153, C2×#393977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301173);