Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394113 |
| 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 | |
3902 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)(6,12)(8,22)(9,10)(13,17)(15,16)(18,32)(19,20)(23,24)(25,29)(27,28)(31,35)(33,34);(5,6,11,16)(7,17,21,27)(8,14,23,30)(9,15,10,12)(13,19,28,20)(18,31,34,35)(22,25,24,29)(26,32,36,33);(2,4)(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,17)(14,26)(15,27)(16,28)(22,32)(23,34)(24,33)(25,31)(29,35)(30,36);(5,8)(6,14)(7,18)(9,24)(10,22)(11,23)(12,25)(13,26)(15,29)(16,30)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36)Polycyclic presentation:
2^9|b^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239639, C2×#394124 |
| maximal normal subgroup of | #72239639, C2×#394124 |
| maximal quotient of | #72249767, C2×#394124, #72316700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301320);