Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14,18,21)(15,22,20,17)(16,19,24,26)(23,28,27,25);(1,3,4,2)(5,7,10,6)(8,11,9,12);(5,8,10,9)(6,11,7,12)(13,15,17,14,18,20,22,21)(16,23,25,19,24,27,28,26);(5,9,10,8)(6,11,7,12)(13,16)(14,19)(15,23)(17,25)(18,24)(20,27)(21,26)(22,28);(16,24)(19,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72240751, #72249595, #72249615 |
| maximal normal subgroup of | #72240751, #72249595, #72249615 |
| maximal quotient of | #72240709, #72240762, #72240763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283647);