Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394022 |
| 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 | |
3969 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)(11,24)(12,26)(15,28)(27,33);(5,6)(7,15)(8,12)(9,21)(10,13)(11,18)(14,23)(16,29)(17,28)(19,26)(20,33)(22,24)(25,30)(27,32)(31,36)(34,35);(2,4)(5,7,10,17)(6,11,13,24)(8,16,19,31)(9,22,23,18)(12,25,26,35)(14,28,21,15)(20,34,32,30)(27,36,33,29);(5,8)(6,12)(7,16)(9,20)(10,19)(11,25)(13,26)(14,27)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(24,35)(28,36)Polycyclic presentation:
2^9|c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279469, #72286993, #72287183, … |
| maximal normal subgroup of | #72279469, #72286993, #72287183, … |
| maximal quotient of | #72281646, #72303554, #72303557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301277);