Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,14)(5,16)(7,18)(8,21)(10,23)(11,25)(13,28)(15,17)(19,31)(22,24)(26,32)(27,29);(1,2)(3,7)(4,8)(5,10)(6,9)(11,26)(12,18)(13,20)(14,21)(15,22)(16,23)(17,24)(19,27)(25,31)(28,30)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,19)(9,18)(10,20)(14,25)(15,27)(16,28)(17,29)(21,32)(22,26)(23,30)(24,31);(1,4)(2,8)(3,11)(5,17)(6,15)(7,19)(9,22)(10,24)(12,27)(13,29)(14,16)(18,26)(20,31)(21,23)(25,28)(30,32)Polycyclic presentation:
2^9|h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#417618, #72237690, #72240435, … |
| maximal normal subgroup of | C2×#417618, #72237690, #72240435, … |
| maximal quotient of | C2×#417618, #72251175, #72251318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324814);