Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16967 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
653 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)(3,7,13,20)(4,8,15,23)(5,16)(6,10)(9,17)(11,18)(12,19,28,30)(14,22,29,24)(21,26)(25,32)(27,31);(1,3)(2,7,10,20)(4,11)(5,12)(6,13)(8,21,23,18)(9,19,16,30)(14,25)(15,26)(17,28)(22,32,24,31)(27,29);(1,4)(2,8,10,23)(3,11)(5,14,17,29)(6,15)(7,18,20,21)(9,24)(12,27,28,25)(13,26)(16,22)(19,31)(30,32);(1,5)(2,9,10,16)(3,12)(4,14,15,29)(6,17)(7,19,20,30)(8,22)(11,25,26,27)(13,28)(18,31)(21,32)(23,24)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318262, #72318278, #72318318, … |
| maximal normal subgroup of | #72318262, #72318278, #72318318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381634);