Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423223 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15313 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(5,6)(10,22)(11,20)(13,28)(14,15)(16,26)(17,18)(21,24)(23,32)(27,30);(1,2,5,8,7,11,6,20)(3,9,14,24,15,25,12,21)(4,16,17,26,19,22,18,10)(13,29,27,32,30,31,28,23);(1,3)(2,9,11,25)(4,13)(5,12)(6,14)(7,15)(8,21,20,24)(10,23,26,32)(16,31,22,29)(17,28)(18,27)(19,30);(1,4)(2,10)(3,13)(5,18)(6,17)(7,19)(8,22)(9,23)(11,26)(12,27)(14,28)(15,30)(16,20)(21,29)(24,31)(25,32)Polycyclic presentation:
2^9|b^2=e,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72213603, #72213704, … |
| maximal normal subgroup of | #72213514, #72213603, #72213704, … |
| maximal quotient of | #72250603, C2×#423223, #72255589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330419);