Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14679 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×H(7,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
16003 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,26,28,25)(14,24)(16,23,32,30)(19,22)(21,27,29,31);(1,3)(2,7)(4,12)(5,19)(6,14)(8,21)(9,24)(10,22)(11,18)(13,20)(15,29)(16,31)(17,28)(23,26)(25,30)(27,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,23)(10,15)(11,25)(13,27)(14,28)(18,30)(19,31)(20,32)(22,29)(24,26);(5,20)(9,18)(11,24)(13,19)(16,32)(23,30)(25,26)(27,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,h^c=hi,e^d=ei89 conjugacy classes — view the full character table.
| maximal subgroup of | #72210464, #72210522, #72210553, … |
| maximal normal subgroup of | #72210464, #72210522, #72210553, … |
| maximal quotient of | #72234504, #72234547, #72234936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299334);