Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14715 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,19)(5,13)(6,11)(7,12)(8,16)(9,20)(10,17)(15,25)(18,24)(21,27)(22,26)(23,31)(28,30)(29,32);(1,3)(2,8)(4,15)(5,17)(6,16)(7,18)(9,25)(10,12)(11,24)(13,14)(19,32)(20,30)(21,31)(22,29)(23,26)(27,28);(1,4)(2,9)(3,15)(5,22)(6,20)(7,21)(8,23)(10,25)(11,27)(12,26)(13,19)(14,28)(16,30)(17,29)(18,31)(24,32);(1,3)(2,10)(4,15)(5,17)(6,16)(7,18)(8,12)(9,25)(11,14)(13,24)(19,32)(20,30)(21,31)(22,29)(23,26)(27,28)Polycyclic presentation:
2^9|e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72215936, #72215958, … |
| maximal normal subgroup of | #72213514, #72215936, #72215958, … |
| maximal quotient of | C2×#394108, #72234091, #72234093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301304);